| L(s) = 1 | − 0.860·2-s + 0.867·3-s − 1.25·4-s − 5-s − 0.746·6-s − 0.263·7-s + 2.80·8-s − 2.24·9-s + 0.860·10-s − 5.32·11-s − 1.09·12-s + 1.19·13-s + 0.227·14-s − 0.867·15-s + 0.105·16-s + 4.46·17-s + 1.93·18-s + 1.25·20-s − 0.228·21-s + 4.57·22-s − 0.920·23-s + 2.43·24-s + 25-s − 1.02·26-s − 4.55·27-s + 0.332·28-s − 9.87·29-s + ⋯ |
| L(s) = 1 | − 0.608·2-s + 0.500·3-s − 0.629·4-s − 0.447·5-s − 0.304·6-s − 0.0997·7-s + 0.991·8-s − 0.749·9-s + 0.272·10-s − 1.60·11-s − 0.315·12-s + 0.331·13-s + 0.0606·14-s − 0.223·15-s + 0.0264·16-s + 1.08·17-s + 0.455·18-s + 0.281·20-s − 0.0499·21-s + 0.976·22-s − 0.192·23-s + 0.496·24-s + 0.200·25-s − 0.201·26-s − 0.875·27-s + 0.0628·28-s − 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7740306218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7740306218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.860T + 2T^{2} \) |
| 3 | \( 1 - 0.867T + 3T^{2} \) |
| 7 | \( 1 + 0.263T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 23 | \( 1 + 0.920T + 23T^{2} \) |
| 29 | \( 1 + 9.87T + 29T^{2} \) |
| 31 | \( 1 + 0.0400T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 8.44T + 47T^{2} \) |
| 53 | \( 1 - 4.20T + 53T^{2} \) |
| 59 | \( 1 + 8.10T + 59T^{2} \) |
| 61 | \( 1 - 0.218T + 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9.91T + 73T^{2} \) |
| 79 | \( 1 - 0.00558T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 + 0.431T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.290289324229784421035443162574, −8.404637608265544935651415666946, −7.76084103469407609255473195323, −7.53877292232525739409332744905, −5.85660221937567279743475293164, −5.31377085915789783518305018121, −4.17870086694896756834650834906, −3.30841797390228575207052584872, −2.28669470065285124084387844963, −0.62629128605433567528969246511,
0.62629128605433567528969246511, 2.28669470065285124084387844963, 3.30841797390228575207052584872, 4.17870086694896756834650834906, 5.31377085915789783518305018121, 5.85660221937567279743475293164, 7.53877292232525739409332744905, 7.76084103469407609255473195323, 8.404637608265544935651415666946, 9.290289324229784421035443162574
