| L(s) = 1 | + 2-s + 3-s + 4-s + 1.83·5-s + 6-s + 7-s + 8-s + 9-s + 1.83·10-s + 1.58·11-s + 12-s + 5.09·13-s + 14-s + 1.83·15-s + 16-s − 0.167·17-s + 18-s + 3.82·19-s + 1.83·20-s + 21-s + 1.58·22-s + 4.24·23-s + 24-s − 1.64·25-s + 5.09·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.819·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.579·10-s + 0.478·11-s + 0.288·12-s + 1.41·13-s + 0.267·14-s + 0.472·15-s + 0.250·16-s − 0.0405·17-s + 0.235·18-s + 0.876·19-s + 0.409·20-s + 0.218·21-s + 0.338·22-s + 0.886·23-s + 0.204·24-s − 0.329·25-s + 0.998·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.145320462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.145320462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.167T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 9.45T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 + 4.31T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 0.727T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 0.624T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65576139754974669054695549202, −7.17126931839333741915805772398, −6.23282326803968188059344344736, −5.80757360047279105306852025946, −5.05889670017883830609128646139, −4.19273057209539973030597768959, −3.51458504799013995286147454852, −2.81414489380097678716747425498, −1.77757671670983430131222939346, −1.23473204040183437487538342426,
1.23473204040183437487538342426, 1.77757671670983430131222939346, 2.81414489380097678716747425498, 3.51458504799013995286147454852, 4.19273057209539973030597768959, 5.05889670017883830609128646139, 5.80757360047279105306852025946, 6.23282326803968188059344344736, 7.17126931839333741915805772398, 7.65576139754974669054695549202
