| L(s) = 1 | + 80·5-s − 486·9-s − 5.76e3·13-s + 2.23e3·17-s − 5.85e4·25-s + 5.06e4·29-s − 2.12e5·37-s + 4.94e4·41-s − 3.88e4·45-s + 1.65e5·49-s − 2.56e5·53-s + 8.09e4·61-s − 4.60e5·65-s − 3.84e4·73-s + 1.77e5·81-s + 1.78e5·85-s − 3.31e6·89-s − 2.46e6·97-s + 4.00e6·101-s − 2.46e6·109-s + 2.79e6·117-s + 1.57e6·121-s − 6.09e6·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.639·5-s − 2/3·9-s − 2.62·13-s + 0.454·17-s − 3.74·25-s + 2.07·29-s − 4.19·37-s + 0.717·41-s − 0.426·45-s + 1.40·49-s − 1.72·53-s + 0.356·61-s − 1.67·65-s − 0.0988·73-s + 1/3·81-s + 0.290·85-s − 4.70·89-s − 2.70·97-s + 3.88·101-s − 1.90·109-s + 1.74·117-s + 0.889·121-s − 3.11·125-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0001399241930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0001399241930\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 p T + p^{6} T^{2} )^{4} \) |
| 7 | $D_4\times C_2$ | \( 1 - 165604 T^{2} + 18091945734 T^{4} - 165604 p^{12} T^{6} + p^{24} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1576516 T^{2} + 319487479206 T^{4} - 1576516 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2880 T + 8554610 T^{2} + 2880 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 1116 T - 2175226 T^{2} - 1116 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 154272388 T^{2} + 10374253923756390 T^{4} - 154272388 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 279579460 T^{2} + 41996156363862342 T^{4} - 279579460 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 25304 T + 321794754 T^{2} - 25304 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2377420708 T^{2} + 2864250173428593990 T^{4} - 2377420708 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 106128 T + 7867925714 T^{2} + 106128 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 24732 T + 9196270886 T^{2} - 24732 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 14783863108 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - 14783863108 p^{12} T^{6} + p^{24} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 11965054468 T^{2} + 35254807719429905286 T^{4} - 11965054468 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 128200 T + 22739407458 T^{2} + 128200 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 127672933828 T^{2} + \)\(72\!\cdots\!58\)\( T^{4} - 127672933828 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 40464 T + 99106782194 T^{2} - 40464 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 65276716028 T^{2} + \)\(16\!\cdots\!10\)\( T^{4} + 65276716028 p^{12} T^{6} + p^{24} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 306730507972 T^{2} + \)\(46\!\cdots\!66\)\( T^{4} - 306730507972 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 19228 T + 154739682726 T^{2} + 19228 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 374724961060 T^{2} + \)\(15\!\cdots\!82\)\( T^{4} - 374724961060 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 849400576132 T^{2} + \)\(37\!\cdots\!78\)\( T^{4} - 849400576132 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1659708 T + 1677544070438 T^{2} + 1659708 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1232420 T + 1634488777158 T^{2} + 1232420 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.24135806637778872604986626234, −6.22037973511594729895242222637, −6.16680140570973856315486226531, −5.65253236282025085043465111414, −5.37085940056014898240592918093, −5.29132945798812208399986519856, −5.29105764543085184207526921210, −4.86890577807709829379306453980, −4.66964869091573280700813532781, −4.27213917793358285713555731398, −4.01865744487114155387028547725, −3.90350369955340376371092034690, −3.55008650128751391938943071632, −3.31286786846642098069004317777, −2.77057739787164855833948852038, −2.73912962020670114910676725592, −2.53900008884672232703702715251, −2.31640163837409264272963748994, −1.77211672189695461379539933619, −1.67683878556332266558099940762, −1.63373687944184562036791185365, −1.05174078842176385310008089776, −0.75046554429102118294499629355, −0.089950355601119815122092482150, −0.00573828573796880464811672721,
0.00573828573796880464811672721, 0.089950355601119815122092482150, 0.75046554429102118294499629355, 1.05174078842176385310008089776, 1.63373687944184562036791185365, 1.67683878556332266558099940762, 1.77211672189695461379539933619, 2.31640163837409264272963748994, 2.53900008884672232703702715251, 2.73912962020670114910676725592, 2.77057739787164855833948852038, 3.31286786846642098069004317777, 3.55008650128751391938943071632, 3.90350369955340376371092034690, 4.01865744487114155387028547725, 4.27213917793358285713555731398, 4.66964869091573280700813532781, 4.86890577807709829379306453980, 5.29105764543085184207526921210, 5.29132945798812208399986519856, 5.37085940056014898240592918093, 5.65253236282025085043465111414, 6.16680140570973856315486226531, 6.22037973511594729895242222637, 6.24135806637778872604986626234
