| L(s) = 1 | + 3.16·2-s − 3·3-s + 2.04·4-s + 5·5-s − 9.50·6-s − 6.36·7-s − 18.8·8-s + 9·9-s + 15.8·10-s + 33.2·11-s − 6.12·12-s + 7.42·13-s − 20.1·14-s − 15·15-s − 76.1·16-s − 80.4·17-s + 28.5·18-s − 77.9·19-s + 10.2·20-s + 19.0·21-s + 105.·22-s − 26.4·23-s + 56.6·24-s + 25·25-s + 23.5·26-s − 27·27-s − 12.9·28-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.577·3-s + 0.255·4-s + 0.447·5-s − 0.646·6-s − 0.343·7-s − 0.834·8-s + 0.333·9-s + 0.501·10-s + 0.910·11-s − 0.147·12-s + 0.158·13-s − 0.385·14-s − 0.258·15-s − 1.19·16-s − 1.14·17-s + 0.373·18-s − 0.941·19-s + 0.114·20-s + 0.198·21-s + 1.01·22-s − 0.239·23-s + 0.481·24-s + 0.200·25-s + 0.177·26-s − 0.192·27-s − 0.0877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.16T + 8T^{2} \) |
| 7 | \( 1 + 6.36T + 343T^{2} \) |
| 11 | \( 1 - 33.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.42T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.4T + 1.21e4T^{2} \) |
| 31 | \( 1 + 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 657.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 642.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 891.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 94.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 268.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 632.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 416.T + 9.12e5T^{2} \) |
| 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
−10.47924714132573858963808201230, −9.344660965565348033409565047938, −8.667650507153327587120821742970, −6.89443125121693639416760394433, −6.32314072898404592104507325340, −5.42158073114155094934730601884, −4.41346485463710854812546846409, −3.53456174271351236701878678208, −1.96849779262704836455072118971, 0,
1.96849779262704836455072118971, 3.53456174271351236701878678208, 4.41346485463710854812546846409, 5.42158073114155094934730601884, 6.32314072898404592104507325340, 6.89443125121693639416760394433, 8.667650507153327587120821742970, 9.344660965565348033409565047938, 10.47924714132573858963808201230
