| L(s) = 1 | + 2.18·2-s + 3-s + 2.79·4-s + 2.18·6-s − 4.37·7-s + 1.73·8-s + 9-s + 2.79·12-s + 4.37·13-s − 9.58·14-s − 1.79·16-s − 3.55·17-s + 2.18·18-s + 5.29·19-s − 4.37·21-s − 8.58·23-s + 1.73·24-s + 9.58·26-s + 27-s − 12.2·28-s + 0.913·29-s − 6.58·31-s − 7.38·32-s − 7.79·34-s + 2.79·36-s − 0.417·37-s + 11.5·38-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.39·4-s + 0.893·6-s − 1.65·7-s + 0.612·8-s + 0.333·9-s + 0.805·12-s + 1.21·13-s − 2.56·14-s − 0.447·16-s − 0.863·17-s + 0.515·18-s + 1.21·19-s − 0.955·21-s − 1.78·23-s + 0.353·24-s + 1.87·26-s + 0.192·27-s − 2.30·28-s + 0.169·29-s − 1.18·31-s − 1.30·32-s − 1.33·34-s + 0.465·36-s − 0.0686·37-s + 1.87·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 - 0.913T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 + 0.417T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 0.913T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 5.10T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
| 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
−7.01960370816487638771998147241, −6.54828798975867937902107333942, −5.94734915777262900377998897074, −5.41182194639252941858617777354, −4.32723838790497135273179430462, −3.74551940284644432756295402596, −3.32610714865974580235987582341, −2.64044484876432207591000354542, −1.66921430310310383203723507868, 0,
1.66921430310310383203723507868, 2.64044484876432207591000354542, 3.32610714865974580235987582341, 3.74551940284644432756295402596, 4.32723838790497135273179430462, 5.41182194639252941858617777354, 5.94734915777262900377998897074, 6.54828798975867937902107333942, 7.01960370816487638771998147241
