L(s) = 1 | + 2·2-s − 9·4-s − 26·5-s − 18·7-s − 16·8-s + 26·9-s − 52·10-s − 36·14-s − 39·16-s + 52·18-s − 238·19-s + 234·20-s + 107·25-s + 162·28-s − 258·32-s + 468·35-s − 234·36-s − 476·38-s + 416·40-s + 1.07e3·41-s − 676·45-s + 908·47-s − 1.14e3·49-s + 214·50-s + 288·56-s − 1.04e3·59-s − 468·63-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 9/8·4-s − 2.32·5-s − 0.971·7-s − 0.707·8-s + 0.962·9-s − 1.64·10-s − 0.687·14-s − 0.609·16-s + 0.680·18-s − 2.87·19-s + 2.61·20-s + 0.855·25-s + 1.09·28-s − 1.42·32-s + 2.26·35-s − 1.08·36-s − 2.03·38-s + 1.64·40-s + 4.10·41-s − 2.23·45-s + 2.81·47-s − 3.34·49-s + 0.605·50-s + 0.687·56-s − 2.29·59-s − 0.935·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 31 | | \( 1 \) |
good | 2 | $D_{4}$ | \( ( 1 - T + 3 p T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 3 | $C_2^2 \wr C_2$ | \( 1 - 26 T^{2} + 1258 T^{4} - 26 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 13 T + 8 p^{2} T^{2} + 13 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 9 T + 696 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 3794 T^{2} + 6968506 T^{4} + 3794 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 1774 T^{2} + 3781618 T^{4} + 1774 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 7296 T^{2} + 39252638 T^{4} + 7296 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 119 T + 16018 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 15104 T^{2} + 398782 p^{2} T^{4} + 15104 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 27326 T^{2} + 692147986 T^{4} + 27326 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 150826 T^{2} + 10602248818 T^{4} + 150826 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 539 T + 179466 T^{2} - 539 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 55494 T^{2} + 8412760298 T^{4} + 55494 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 454 T + 257166 T^{2} - 454 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 372474 T^{2} + 78680620658 T^{4} + 372474 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 521 T + 378158 T^{2} + 521 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 83346 T^{2} + 96781517426 T^{4} - 83346 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 464 T + 442806 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 315 T + 553822 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 688084 T^{2} + 420958406278 T^{4} + 688084 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 980536 T^{2} + 710609774766 T^{4} + 980536 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 1448594 T^{2} + 1171132620538 T^{4} + 1448594 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 2198696 T^{2} + 2176677610926 T^{4} + 2198696 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 631 T + 1886746 T^{2} + 631 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.02046290631354647859595877353, −6.92937868592149125917192000464, −6.91504930514003897260303208172, −6.60135369866178129102396369405, −6.44490213609479494397664412437, −6.00430589599240144006956125484, −5.83070411006485904836602730403, −5.71372933501465674669649525135, −5.33202687935029027930677225155, −5.01887963405833409699993558750, −4.61348168667065878553313987071, −4.46625427874135165446456340178, −4.41659905955214649197242286001, −4.18715134685488972937009626463, −4.02920508322069915214020911689, −3.80138904904494507834598153563, −3.71608551008760597198003859656, −3.37103931989075782670582257504, −2.97430794323966977899274207757, −2.44915939111975170774729494252, −2.31521314016812585627955220389, −2.29367477706830516547467002046, −1.53682903760369571976798832900, −1.15687922730036050423664898996, −0.887457176185676670511654897778, 0, 0, 0, 0,
0.887457176185676670511654897778, 1.15687922730036050423664898996, 1.53682903760369571976798832900, 2.29367477706830516547467002046, 2.31521314016812585627955220389, 2.44915939111975170774729494252, 2.97430794323966977899274207757, 3.37103931989075782670582257504, 3.71608551008760597198003859656, 3.80138904904494507834598153563, 4.02920508322069915214020911689, 4.18715134685488972937009626463, 4.41659905955214649197242286001, 4.46625427874135165446456340178, 4.61348168667065878553313987071, 5.01887963405833409699993558750, 5.33202687935029027930677225155, 5.71372933501465674669649525135, 5.83070411006485904836602730403, 6.00430589599240144006956125484, 6.44490213609479494397664412437, 6.60135369866178129102396369405, 6.91504930514003897260303208172, 6.92937868592149125917192000464, 7.02046290631354647859595877353