L(s) = 1 | + 16·4-s + 52·7-s + 292·13-s + 64·16-s + 1.48e3·19-s − 2.03e3·25-s + 832·28-s + 5.05e3·31-s + 8.84e3·37-s − 428·43-s + 1.01e4·49-s + 4.67e3·52-s − 4.49e3·61-s − 1.02e3·64-s − 1.64e4·67-s − 4.16e4·73-s + 2.36e4·76-s + 1.67e4·79-s + 1.51e4·91-s + 3.80e3·97-s − 3.25e4·100-s − 1.13e4·103-s + 1.82e4·109-s + 3.32e3·112-s − 4.59e4·121-s + 8.08e4·124-s + 127-s + ⋯ |
L(s) = 1 | + 4-s + 1.06·7-s + 1.72·13-s + 1/4·16-s + 4.09·19-s − 3.25·25-s + 1.06·28-s + 5.26·31-s + 6.46·37-s − 0.231·43-s + 4.21·49-s + 1.72·52-s − 1.20·61-s − 1/4·64-s − 3.66·67-s − 7.80·73-s + 4.09·76-s + 2.68·79-s + 1.83·91-s + 0.404·97-s − 3.25·100-s − 1.07·103-s + 1.53·109-s + 0.265·112-s − 3.13·121-s + 5.26·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(39.79573656\) |
\(L(\frac12)\) |
\(\approx\) |
\(39.79573656\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2032 T^{2} + 2370193 T^{4} + 1986444592 T^{6} + 1327998595936 T^{8} + 1986444592 p^{8} T^{10} + 2370193 p^{16} T^{12} + 2032 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 - 26 T - 4052 T^{2} + 1924 T^{3} + 14966107 T^{4} + 1924 p^{4} T^{5} - 4052 p^{8} T^{6} - 26 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 + 45928 T^{2} + 1164655534 T^{4} + 195861241984 p^{2} T^{6} + 26937520230835 p^{4} T^{8} + 195861241984 p^{10} T^{10} + 1164655534 p^{16} T^{12} + 45928 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 146 T - 1295 p T^{2} + 2769766 T^{3} + 30961804 T^{4} + 2769766 p^{4} T^{5} - 1295 p^{9} T^{6} - 146 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 104 T^{2} + 13445531079 T^{4} + 104 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 - 370 T + 61344 T^{2} - 370 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( 1 + 657664 T^{2} + 196113002110 T^{4} + 52473014039412736 T^{6} + \)\(15\!\cdots\!59\)\( T^{8} + 52473014039412736 p^{8} T^{10} + 196113002110 p^{16} T^{12} + 657664 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 + 1983160 T^{2} + 1949268379801 T^{4} + 1949768305276871320 T^{6} + \)\(17\!\cdots\!80\)\( T^{8} + 1949768305276871320 p^{8} T^{10} + 1949268379801 p^{16} T^{12} + 1983160 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 - 2528 T + 3024778 T^{2} - 3839940992 T^{3} + 4575082124131 T^{4} - 3839940992 p^{4} T^{5} + 3024778 p^{8} T^{6} - 2528 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2212 T + 3297531 T^{2} - 2212 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 41 | \( 1 + 954232 T^{2} - 14906791774802 T^{4} - 145520354823542912 T^{6} + \)\(17\!\cdots\!31\)\( T^{8} - 145520354823542912 p^{8} T^{10} - 14906791774802 p^{16} T^{12} + 954232 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 214 T - 1972172 T^{2} - 1031401676 T^{3} - 7772572839173 T^{4} - 1031401676 p^{4} T^{5} - 1972172 p^{8} T^{6} + 214 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 + 5282452 T^{2} - 17108794424102 T^{4} - 13784451614761448432 T^{6} + \)\(79\!\cdots\!71\)\( T^{8} - 13784451614761448432 p^{8} T^{10} - 17108794424102 p^{16} T^{12} + 5282452 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 14240632 T^{2} + 170431500573906 T^{4} - 14240632 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 + 30153076 T^{2} + 389575067804890 T^{4} + \)\(68\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!59\)\( T^{8} + \)\(68\!\cdots\!44\)\( p^{8} T^{10} + 389575067804890 p^{16} T^{12} + 30153076 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 + 2248 T - 15221351 T^{2} - 16673027096 T^{3} + 149392419396880 T^{4} - 16673027096 p^{4} T^{5} - 15221351 p^{8} T^{6} + 2248 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8218 T + 14393164 T^{2} + 105520089724 T^{3} + 1057126232890555 T^{4} + 105520089724 p^{4} T^{5} + 14393164 p^{8} T^{6} + 8218 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 94710496 T^{2} + 700726549794 p^{2} T^{4} - 94710496 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 + 10400 T + 83729319 T^{2} + 10400 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 79 | \( ( 1 - 8366 T + 21423508 T^{2} + 245405851324 T^{3} - 1884523393956293 T^{4} + 245405851324 p^{4} T^{5} + 21423508 p^{8} T^{6} - 8366 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 + 33630580 T^{2} + 874409024368426 T^{4} - \)\(14\!\cdots\!40\)\( T^{6} - \)\(75\!\cdots\!05\)\( T^{8} - \)\(14\!\cdots\!40\)\( p^{8} T^{10} + 874409024368426 p^{16} T^{12} + 33630580 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 - 216095728 T^{2} + 19466598550909983 T^{4} - 216095728 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 1904 T - 31757942 T^{2} + 269749969216 T^{3} - 6906403490273693 T^{4} + 269749969216 p^{4} T^{5} - 31757942 p^{8} T^{6} - 1904 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.26662449316430700267579345150, −4.66335949571504402211868407789, −4.64383919503614824131568610289, −4.59739660038598757574549495486, −4.56847508996826725205346763203, −4.28055886920367386274674355089, −4.05294618408883440115428559633, −4.01317155604847908618448620570, −3.91863388345103283503726769450, −3.59193128112342594486537391180, −3.05928945256645453273386583676, −3.05586336405602596087522997331, −2.86169762208314961129023886427, −2.82415417122169905842604623679, −2.64774122926893914925045338408, −2.64354249711713883438601794027, −2.17422588301308087890218105363, −1.77675778657512717655283130357, −1.50344027871504741698269651573, −1.49719463420111453654086741640, −1.25283496357329835433936264243, −0.926994359507860765101763590810, −0.798914663529718275847047179341, −0.72772551199975859222039242619, −0.34511103251273190100850181268,
0.34511103251273190100850181268, 0.72772551199975859222039242619, 0.798914663529718275847047179341, 0.926994359507860765101763590810, 1.25283496357329835433936264243, 1.49719463420111453654086741640, 1.50344027871504741698269651573, 1.77675778657512717655283130357, 2.17422588301308087890218105363, 2.64354249711713883438601794027, 2.64774122926893914925045338408, 2.82415417122169905842604623679, 2.86169762208314961129023886427, 3.05586336405602596087522997331, 3.05928945256645453273386583676, 3.59193128112342594486537391180, 3.91863388345103283503726769450, 4.01317155604847908618448620570, 4.05294618408883440115428559633, 4.28055886920367386274674355089, 4.56847508996826725205346763203, 4.59739660038598757574549495486, 4.64383919503614824131568610289, 4.66335949571504402211868407789, 5.26662449316430700267579345150
Plot not available for L-functions of degree greater than 10.