| L(s) = 1 | − 1.00·2-s − 3-s − 0.989·4-s − 2.22·5-s + 1.00·6-s + 1.19·7-s + 3.00·8-s + 9-s + 2.23·10-s + 0.183·11-s + 0.989·12-s − 0.922·13-s − 1.20·14-s + 2.22·15-s − 1.04·16-s − 3.54·17-s − 1.00·18-s − 1.94·19-s + 2.20·20-s − 1.19·21-s − 0.184·22-s + 1.99·23-s − 3.00·24-s − 0.0487·25-s + 0.926·26-s − 27-s − 1.18·28-s + ⋯ |
| L(s) = 1 | − 0.710·2-s − 0.577·3-s − 0.494·4-s − 0.995·5-s + 0.410·6-s + 0.452·7-s + 1.06·8-s + 0.333·9-s + 0.707·10-s + 0.0552·11-s + 0.285·12-s − 0.255·13-s − 0.321·14-s + 0.574·15-s − 0.260·16-s − 0.859·17-s − 0.236·18-s − 0.446·19-s + 0.492·20-s − 0.260·21-s − 0.0392·22-s + 0.416·23-s − 0.613·24-s − 0.00975·25-s + 0.181·26-s − 0.192·27-s − 0.223·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4590053655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4590053655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 677 | \( 1 - T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 - 0.183T + 11T^{2} \) |
| 13 | \( 1 + 0.922T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 - 1.99T + 23T^{2} \) |
| 29 | \( 1 + 0.968T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 0.764T + 71T^{2} \) |
| 73 | \( 1 - 0.600T + 73T^{2} \) |
| 79 | \( 1 - 4.72T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.021951375823544971718239664954, −8.384206113761710695141073120138, −7.71335968204368327166656636202, −7.04121330297774901917748286707, −6.05632931733176937166822142468, −4.73974051881417445107807726918, −4.56854588711104774909513175097, −3.44005301390165919704142984086, −1.84116029761206790002332342583, −0.51987472121065890675482098755,
0.51987472121065890675482098755, 1.84116029761206790002332342583, 3.44005301390165919704142984086, 4.56854588711104774909513175097, 4.73974051881417445107807726918, 6.05632931733176937166822142468, 7.04121330297774901917748286707, 7.71335968204368327166656636202, 8.384206113761710695141073120138, 9.021951375823544971718239664954
