| L(s) = 1 | − 2-s − 2·4-s + 3·5-s − 5·7-s + 3·8-s − 3·10-s − 2·11-s − 3·13-s + 5·14-s + 16-s − 5·17-s − 4·19-s − 6·20-s + 2·22-s − 8·23-s + 3·26-s + 10·28-s + 5·29-s − 6·31-s − 2·32-s + 5·34-s − 15·35-s − 13·37-s + 4·38-s + 9·40-s + 9·41-s − 12·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s + 1.34·5-s − 1.88·7-s + 1.06·8-s − 0.948·10-s − 0.603·11-s − 0.832·13-s + 1.33·14-s + 1/4·16-s − 1.21·17-s − 0.917·19-s − 1.34·20-s + 0.426·22-s − 1.66·23-s + 0.588·26-s + 1.88·28-s + 0.928·29-s − 1.07·31-s − 0.353·32-s + 0.857·34-s − 2.53·35-s − 2.13·37-s + 0.648·38-s + 1.42·40-s + 1.40·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 181 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 17 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 39 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 24 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T - 44 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 131 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 17 T + 188 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 163 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 105 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 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
−13.1260906313, −12.9654686087, −12.5340929837, −12.1328302328, −11.6195135500, −10.8417221175, −10.3279388415, −10.1837782000, −9.88298443262, −9.56890780292, −8.99460916802, −8.86457212906, −8.44781029605, −7.69910168004, −7.34151316221, −6.61818534297, −6.23998173161, −6.07010671379, −5.22719379317, −4.96422131168, −4.15487523700, −3.78484194907, −2.93938189596, −2.30710118592, −1.77306099285, 0, 0,
1.77306099285, 2.30710118592, 2.93938189596, 3.78484194907, 4.15487523700, 4.96422131168, 5.22719379317, 6.07010671379, 6.23998173161, 6.61818534297, 7.34151316221, 7.69910168004, 8.44781029605, 8.86457212906, 8.99460916802, 9.56890780292, 9.88298443262, 10.1837782000, 10.3279388415, 10.8417221175, 11.6195135500, 12.1328302328, 12.5340929837, 12.9654686087, 13.1260906313
