L(s) = 1 | + 2-s − 0.617·3-s + 4-s − 3.87·5-s − 0.617·6-s − 1.63·7-s + 8-s − 2.61·9-s − 3.87·10-s + 0.568·11-s − 0.617·12-s − 1.82·13-s − 1.63·14-s + 2.39·15-s + 16-s + 4.77·17-s − 2.61·18-s − 19-s − 3.87·20-s + 1.01·21-s + 0.568·22-s + 2.24·23-s − 0.617·24-s + 10.0·25-s − 1.82·26-s + 3.46·27-s − 1.63·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s − 1.73·5-s − 0.251·6-s − 0.619·7-s + 0.353·8-s − 0.873·9-s − 1.22·10-s + 0.171·11-s − 0.178·12-s − 0.506·13-s − 0.437·14-s + 0.617·15-s + 0.250·16-s + 1.15·17-s − 0.617·18-s − 0.229·19-s − 0.866·20-s + 0.220·21-s + 0.121·22-s + 0.467·23-s − 0.125·24-s + 2.00·25-s − 0.358·26-s + 0.667·27-s − 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 + 0.617T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 0.568T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 + 9.56T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 + 1.35T + 61T^{2} \) |
| 67 | \( 1 + 4.19T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 0.437T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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
−7.47758761347178725394868052614, −6.63660055517529078986839511443, −6.22891607083881094885364841050, −5.17289216277331227308805982077, −4.71453569514342817481647922262, −3.87661771183912049932638834362, −3.16518199050292747047136040582, −2.75030039516968942052805827852, −1.03343201681354423294145119237, 0,
1.03343201681354423294145119237, 2.75030039516968942052805827852, 3.16518199050292747047136040582, 3.87661771183912049932638834362, 4.71453569514342817481647922262, 5.17289216277331227308805982077, 6.22891607083881094885364841050, 6.63660055517529078986839511443, 7.47758761347178725394868052614