| L(s) = 1 | − 0.0234·2-s + 3-s − 1.99·4-s − 0.565·5-s − 0.0234·6-s + 3.64·7-s + 0.0937·8-s + 9-s + 0.0132·10-s − 1.09·11-s − 1.99·12-s − 1.94·13-s − 0.0854·14-s − 0.565·15-s + 3.99·16-s + 1.32·17-s − 0.0234·18-s + 4.05·19-s + 1.13·20-s + 3.64·21-s + 0.0256·22-s + 6.02·23-s + 0.0937·24-s − 4.67·25-s + 0.0455·26-s + 27-s − 7.29·28-s + ⋯ |
| L(s) = 1 | − 0.0165·2-s + 0.577·3-s − 0.999·4-s − 0.253·5-s − 0.00956·6-s + 1.37·7-s + 0.0331·8-s + 0.333·9-s + 0.00419·10-s − 0.330·11-s − 0.577·12-s − 0.539·13-s − 0.0228·14-s − 0.146·15-s + 0.999·16-s + 0.322·17-s − 0.00552·18-s + 0.931·19-s + 0.253·20-s + 0.795·21-s + 0.00547·22-s + 1.25·23-s + 0.0191·24-s − 0.935·25-s + 0.00894·26-s + 0.192·27-s − 1.37·28-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
| \(L(1)\) |
\(\approx\) |
\(1.887576271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.887576271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0234T + 2T^{2} \) |
| 5 | \( 1 + 0.565T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 7.96T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 + 4.38T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 0.100T + 79T^{2} \) |
| 83 | \( 1 - 0.272T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
| 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
−9.287079830587473141831599461242, −8.249514054770137028214077613917, −7.76522042173893729204538754878, −7.21567474070990020871137408227, −5.56253989024669318108501786990, −5.10502271796216626021802980783, −4.24399337692123222277455744038, −3.43527053738734828302861668508, −2.18759623247979309891673806144, −0.942329883396067221723866334276,
0.942329883396067221723866334276, 2.18759623247979309891673806144, 3.43527053738734828302861668508, 4.24399337692123222277455744038, 5.10502271796216626021802980783, 5.56253989024669318108501786990, 7.21567474070990020871137408227, 7.76522042173893729204538754878, 8.249514054770137028214077613917, 9.287079830587473141831599461242
