| L(s) = 1 | − 4·2-s − 6·3-s + 16·4-s + 24·6-s + 118·7-s − 64·8-s − 207·9-s + 192·11-s − 96·12-s − 1.10e3·13-s − 472·14-s + 256·16-s − 762·17-s + 828·18-s − 2.74e3·19-s − 708·21-s − 768·22-s − 1.56e3·23-s + 384·24-s + 4.42e3·26-s + 2.70e3·27-s + 1.88e3·28-s + 5.91e3·29-s − 6.86e3·31-s − 1.02e3·32-s − 1.15e3·33-s + 3.04e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.272·6-s + 0.910·7-s − 0.353·8-s − 0.851·9-s + 0.478·11-s − 0.192·12-s − 1.81·13-s − 0.643·14-s + 1/4·16-s − 0.639·17-s + 0.602·18-s − 1.74·19-s − 0.350·21-s − 0.338·22-s − 0.617·23-s + 0.136·24-s + 1.28·26-s + 0.712·27-s + 0.455·28-s + 1.30·29-s − 1.28·31-s − 0.176·32-s − 0.184·33-s + 0.452·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 118 T + p^{5} T^{2} \) |
| 11 | \( 1 - 192 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 + 762 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2740 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1566 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5910 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6868 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 41 | \( 1 + 378 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2434 T + p^{5} T^{2} \) |
| 47 | \( 1 + 13122 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9174 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34980 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9838 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33722 T + p^{5} T^{2} \) |
| 71 | \( 1 - 70212 T + p^{5} T^{2} \) |
| 73 | \( 1 + 21986 T + p^{5} T^{2} \) |
| 79 | \( 1 - 4520 T + p^{5} T^{2} \) |
| 83 | \( 1 - 109074 T + p^{5} T^{2} \) |
| 89 | \( 1 - 38490 T + p^{5} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.33656242595144276128435263938, −12.44009490158490100377502539167, −11.48549207111135330828882525646, −10.47829637350612823948653778647, −9.031718596368505927690264244494, −7.895514057346205947945790234940, −6.42222539135573081420566338087, −4.76941136820516591871315391426, −2.21759630580446080702429995502, 0,
2.21759630580446080702429995502, 4.76941136820516591871315391426, 6.42222539135573081420566338087, 7.895514057346205947945790234940, 9.031718596368505927690264244494, 10.47829637350612823948653778647, 11.48549207111135330828882525646, 12.44009490158490100377502539167, 14.33656242595144276128435263938
