| L(s) = 1 | + 1.48·2-s − 0.806·3-s + 0.193·4-s − 1.19·6-s − 3.35·7-s − 2.67·8-s − 2.35·9-s + 0.962·11-s − 0.156·12-s − 6.15·13-s − 4.96·14-s − 4.35·16-s + 6.31·17-s − 3.48·18-s − 19-s + 2.70·21-s + 1.42·22-s + 4.96·23-s + 2.15·24-s − 9.11·26-s + 4.31·27-s − 0.649·28-s − 3.61·29-s − 5.92·31-s − 1.09·32-s − 0.775·33-s + 9.35·34-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.465·3-s + 0.0969·4-s − 0.487·6-s − 1.26·7-s − 0.945·8-s − 0.783·9-s + 0.290·11-s − 0.0451·12-s − 1.70·13-s − 1.32·14-s − 1.08·16-s + 1.53·17-s − 0.820·18-s − 0.229·19-s + 0.589·21-s + 0.303·22-s + 1.03·23-s + 0.440·24-s − 1.78·26-s + 0.829·27-s − 0.122·28-s − 0.670·29-s − 1.06·31-s − 0.193·32-s − 0.135·33-s + 1.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 + 0.387T + 73T^{2} \) |
| 79 | \( 1 + 0.836T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 - 7.08T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−10.68706585680744218921860629453, −9.596030474136190821664917785490, −9.066418741591584822201264381691, −7.52786815767506675418148855799, −6.53381958851570359278476920409, −5.61833942145190438759590697906, −4.99171169998799150443949090789, −3.58462042181120631379624837386, −2.81151268124156327008609849566, 0,
2.81151268124156327008609849566, 3.58462042181120631379624837386, 4.99171169998799150443949090789, 5.61833942145190438759590697906, 6.53381958851570359278476920409, 7.52786815767506675418148855799, 9.066418741591584822201264381691, 9.596030474136190821664917785490, 10.68706585680744218921860629453
