| L(s) = 1 | + 4·2-s + 18·3-s − 20·4-s + 72·6-s + 98·7-s − 96·8-s + 243·9-s − 176·11-s − 360·12-s − 692·13-s + 392·14-s − 176·16-s + 428·17-s + 972·18-s − 1.82e3·19-s + 1.76e3·21-s − 704·22-s − 8.03e3·23-s − 1.72e3·24-s − 2.76e3·26-s + 2.91e3·27-s − 1.96e3·28-s − 2.94e3·29-s + 1.47e4·31-s − 3.90e3·32-s − 3.16e3·33-s + 1.71e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 5/8·4-s + 0.816·6-s + 0.755·7-s − 0.530·8-s + 9-s − 0.438·11-s − 0.721·12-s − 1.13·13-s + 0.534·14-s − 0.171·16-s + 0.359·17-s + 0.707·18-s − 1.15·19-s + 0.872·21-s − 0.310·22-s − 3.16·23-s − 0.612·24-s − 0.803·26-s + 0.769·27-s − 0.472·28-s − 0.650·29-s + 2.75·31-s − 0.673·32-s − 0.506·33-s + 0.253·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{2} T + 9 p^{2} T^{2} - p^{7} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 16 p T + 9846 T^{2} + 16 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 692 T + 856894 T^{2} + 692 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 428 T + 1236582 T^{2} - 428 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 96 p T + 5676294 T^{2} + 96 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8032 T + 1215330 p T^{2} + 8032 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2948 T + 32688446 T^{2} + 2948 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14760 T + 109231790 T^{2} - 14760 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 156 T - 37875634 T^{2} + 156 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5980 T + 145983702 T^{2} + 5980 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 25672 T + 418423654 T^{2} + 25672 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21904 T + 571171070 T^{2} + 21904 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 53948 T + 1562783310 T^{2} - 53948 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 26296 T + 547660454 T^{2} - 26296 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16788 T + 1437884126 T^{2} + 16788 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16264 T + 2696002390 T^{2} + 16264 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22264 T + 3731796926 T^{2} - 22264 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 28684 T + 3989062102 T^{2} + 28684 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 64368 T + 3610665822 T^{2} + 64368 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 75848 T + 5983113110 T^{2} + 75848 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 32964 T + 2468206070 T^{2} - 32964 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 242604 T + 31399386246 T^{2} + 242604 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.866634865040001833968207904119, −9.624098384703269709008794717098, −8.640954655578451520341891847188, −8.459936643822465807366317583416, −8.179210618175871646417052096047, −7.84645857362853558472651039460, −7.14691476276241452517742846462, −6.73260820552851395612885382569, −6.07108694390052650346215877836, −5.47411936966908682453995990104, −4.98777858190430365580971602859, −4.48541764726733416579475793970, −4.10611006086103991160756584351, −3.81270405039926814094261874892, −2.92559776217930425644507863125, −2.40100739326740421521271948501, −1.96297649682581384422844495842, −1.32798582588695761971717201510, 0, 0,
1.32798582588695761971717201510, 1.96297649682581384422844495842, 2.40100739326740421521271948501, 2.92559776217930425644507863125, 3.81270405039926814094261874892, 4.10611006086103991160756584351, 4.48541764726733416579475793970, 4.98777858190430365580971602859, 5.47411936966908682453995990104, 6.07108694390052650346215877836, 6.73260820552851395612885382569, 7.14691476276241452517742846462, 7.84645857362853558472651039460, 8.179210618175871646417052096047, 8.459936643822465807366317583416, 8.640954655578451520341891847188, 9.624098384703269709008794717098, 9.866634865040001833968207904119
