L(s) = 1 | + 128·13-s − 3.87e3·25-s + 3.58e3·37-s − 2.74e3·49-s + 1.19e4·61-s − 2.40e4·73-s − 3.91e4·97-s + 6.41e4·109-s + 1.09e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.15e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 0.757·13-s − 6.19·25-s + 2.61·37-s − 8/7·49-s + 3.19·61-s − 4.50·73-s − 4.15·97-s + 5.40·109-s + 7.48·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 7.53·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8530285383\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8530285383\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p^{3} T^{2} )^{8} \) |
good | 5 | \( ( 1 + 1936 T^{2} + 500204 p T^{4} + 2287180912 T^{6} + 1603004322886 T^{8} + 2287180912 p^{8} T^{10} + 500204 p^{17} T^{12} + 1936 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 11 | \( ( 1 - 54784 T^{2} + 1493191996 T^{4} - 26274699394816 T^{6} + 392638164053397766 T^{8} - 26274699394816 p^{8} T^{10} + 1493191996 p^{16} T^{12} - 54784 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 13 | \( ( 1 - 32 T + 56380 T^{2} - 1811936 T^{3} + 1665416134 T^{4} - 1811936 p^{4} T^{5} + 56380 p^{8} T^{6} - 32 p^{12} T^{7} + p^{16} T^{8} )^{4} \) |
| 17 | \( ( 1 + 335248 T^{2} + 62211910684 T^{4} + 7722867903851632 T^{6} + \)\(73\!\cdots\!50\)\( T^{8} + 7722867903851632 p^{8} T^{10} + 62211910684 p^{16} T^{12} + 335248 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 19 | \( ( 1 - 791848 T^{2} + 296786911324 T^{4} - 68884174829971480 T^{6} + \)\(10\!\cdots\!86\)\( T^{8} - 68884174829971480 p^{8} T^{10} + 296786911324 p^{16} T^{12} - 791848 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1115584 T^{2} + 620079367996 T^{4} - 229866578247638848 T^{6} + \)\(68\!\cdots\!90\)\( T^{8} - 229866578247638848 p^{8} T^{10} + 620079367996 p^{16} T^{12} - 1115584 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 29 | \( ( 1 + 3092392 T^{2} + 5000167429468 T^{4} + 5556600318831192472 T^{6} + \)\(45\!\cdots\!74\)\( T^{8} + 5556600318831192472 p^{8} T^{10} + 5000167429468 p^{16} T^{12} + 3092392 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 31 | \( ( 1 - 3394856 T^{2} + 6301236603868 T^{4} - 8657196664558293272 T^{6} + \)\(91\!\cdots\!74\)\( T^{8} - 8657196664558293272 p^{8} T^{10} + 6301236603868 p^{16} T^{12} - 3394856 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 37 | \( ( 1 - 896 T + 2637436 T^{2} - 531934592 T^{3} + 5277277983430 T^{4} - 531934592 p^{4} T^{5} + 2637436 p^{8} T^{6} - 896 p^{12} T^{7} + p^{16} T^{8} )^{4} \) |
| 41 | \( ( 1 + 11561104 T^{2} + 75268509939484 T^{4} + \)\(33\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{8} + \)\(33\!\cdots\!08\)\( p^{8} T^{10} + 75268509939484 p^{16} T^{12} + 11561104 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 43 | \( ( 1 - 11555464 T^{2} + 90501291016348 T^{4} - \)\(47\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!94\)\( T^{8} - \)\(47\!\cdots\!32\)\( p^{8} T^{10} + 90501291016348 p^{16} T^{12} - 11555464 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 47 | \( ( 1 - 25192424 T^{2} + 315812513879644 T^{4} - \)\(25\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!90\)\( T^{8} - \)\(25\!\cdots\!36\)\( p^{8} T^{10} + 315812513879644 p^{16} T^{12} - 25192424 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 53 | \( ( 1 + 31129096 T^{2} + 556357716250012 T^{4} + \)\(68\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!82\)\( T^{8} + \)\(68\!\cdots\!76\)\( p^{8} T^{10} + 556357716250012 p^{16} T^{12} + 31129096 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 59 | \( ( 1 - 42402664 T^{2} + 957249556313308 T^{4} - \)\(16\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!34\)\( T^{8} - \)\(16\!\cdots\!60\)\( p^{8} T^{10} + 957249556313308 p^{16} T^{12} - 42402664 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 61 | \( ( 1 - 2976 T + 30146620 T^{2} - 49614869088 T^{3} + 494830549535430 T^{4} - 49614869088 p^{4} T^{5} + 30146620 p^{8} T^{6} - 2976 p^{12} T^{7} + p^{16} T^{8} )^{4} \) |
| 67 | \( ( 1 - 88955912 T^{2} + 4323088836599452 T^{4} - \)\(14\!\cdots\!08\)\( T^{6} + \)\(33\!\cdots\!74\)\( T^{8} - \)\(14\!\cdots\!08\)\( p^{8} T^{10} + 4323088836599452 p^{16} T^{12} - 88955912 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 71 | \( ( 1 - 137966080 T^{2} + 8490085602160060 T^{4} - \)\(32\!\cdots\!32\)\( T^{6} + \)\(91\!\cdots\!78\)\( T^{8} - \)\(32\!\cdots\!32\)\( p^{8} T^{10} + 8490085602160060 p^{16} T^{12} - 137966080 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 73 | \( ( 1 + 6008 T + 101184124 T^{2} + 406739815880 T^{3} + 4017310035289606 T^{4} + 406739815880 p^{4} T^{5} + 101184124 p^{8} T^{6} + 6008 p^{12} T^{7} + p^{16} T^{8} )^{4} \) |
| 79 | \( ( 1 - 267664520 T^{2} + 32399897228583196 T^{4} - \)\(23\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!40\)\( p^{8} T^{10} + 32399897228583196 p^{16} T^{12} - 267664520 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 83 | \( ( 1 - 244012552 T^{2} + 30440472310890652 T^{4} - \)\(24\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!62\)\( T^{8} - \)\(24\!\cdots\!44\)\( p^{8} T^{10} + 30440472310890652 p^{16} T^{12} - 244012552 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 89 | \( ( 1 + 42631312 T^{2} + 6283562299988764 T^{4} + \)\(96\!\cdots\!32\)\( T^{6} + \)\(21\!\cdots\!38\)\( T^{8} + \)\(96\!\cdots\!32\)\( p^{8} T^{10} + 6283562299988764 p^{16} T^{12} + 42631312 p^{24} T^{14} + p^{32} T^{16} )^{2} \) |
| 97 | \( ( 1 + 9784 T + 265382524 T^{2} + 1933211397640 T^{3} + 30401418939548038 T^{4} + 1933211397640 p^{4} T^{5} + 265382524 p^{8} T^{6} + 9784 p^{12} T^{7} + p^{16} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.00895908079186179574381359162, −1.93289291031841054547324099926, −1.74103287292888160508054556548, −1.73989927315449984891452095382, −1.64971805857806836591155042335, −1.63699037278669383501003881752, −1.59626424623075414515106534086, −1.54671843332345396071331270346, −1.29548039016617707815532907012, −1.21465184243156041741082646513, −1.20082628343300595676539070061, −1.18614989633699163264883603414, −1.05056799132918014547158909156, −0.962015983649229730072511941605, −0.853555040850456735183163907807, −0.847022011818736250010660321098, −0.61949663281821872712598069616, −0.61848394904991492506410200471, −0.53028631887705231849946409165, −0.44387502691704536842942004899, −0.39201756808621214396138989770, −0.28014738424339353518724077617, −0.23790791237874339738429970903, −0.07180544888651171462007560825, −0.05022777185488664973294367242,
0.05022777185488664973294367242, 0.07180544888651171462007560825, 0.23790791237874339738429970903, 0.28014738424339353518724077617, 0.39201756808621214396138989770, 0.44387502691704536842942004899, 0.53028631887705231849946409165, 0.61848394904991492506410200471, 0.61949663281821872712598069616, 0.847022011818736250010660321098, 0.853555040850456735183163907807, 0.962015983649229730072511941605, 1.05056799132918014547158909156, 1.18614989633699163264883603414, 1.20082628343300595676539070061, 1.21465184243156041741082646513, 1.29548039016617707815532907012, 1.54671843332345396071331270346, 1.59626424623075414515106534086, 1.63699037278669383501003881752, 1.64971805857806836591155042335, 1.73989927315449984891452095382, 1.74103287292888160508054556548, 1.93289291031841054547324099926, 2.00895908079186179574381359162
Plot not available for L-functions of degree greater than 10.