| L(s) = 1 | − 0.581·2-s − 2.46·3-s − 1.66·4-s + 1.43·6-s − 1.67·7-s + 2.12·8-s + 3.07·9-s + 5.38·11-s + 4.09·12-s − 0.325·13-s + 0.972·14-s + 2.08·16-s − 0.627·17-s − 1.78·18-s − 1.81·19-s + 4.12·21-s − 3.12·22-s + 0.558·23-s − 5.24·24-s + 0.189·26-s − 0.196·27-s + 2.78·28-s + 1.60·29-s + 31-s − 5.46·32-s − 13.2·33-s + 0.364·34-s + ⋯ |
| L(s) = 1 | − 0.410·2-s − 1.42·3-s − 0.831·4-s + 0.584·6-s − 0.632·7-s + 0.752·8-s + 1.02·9-s + 1.62·11-s + 1.18·12-s − 0.0903·13-s + 0.259·14-s + 0.522·16-s − 0.152·17-s − 0.421·18-s − 0.415·19-s + 0.900·21-s − 0.666·22-s + 0.116·23-s − 1.07·24-s + 0.0371·26-s − 0.0378·27-s + 0.525·28-s + 0.297·29-s + 0.179·31-s − 0.966·32-s − 2.31·33-s + 0.0625·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.581T + 2T^{2} \) |
| 3 | \( 1 + 2.46T + 3T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 0.325T + 13T^{2} \) |
| 17 | \( 1 + 0.627T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 0.558T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 9.93T + 47T^{2} \) |
| 53 | \( 1 + 3.51T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.69T + 61T^{2} \) |
| 67 | \( 1 - 0.686T + 67T^{2} \) |
| 71 | \( 1 - 1.00T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
| 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
−9.968749540253452170107207438533, −9.139243844496448643502363350997, −8.395287926742891746683400570219, −6.95512054653585480974233314849, −6.45136232710592368889883233471, −5.42510254810447795726185271561, −4.54150210189823859099802798523, −3.59720404918672105320782946095, −1.34820977695547805735406548244, 0,
1.34820977695547805735406548244, 3.59720404918672105320782946095, 4.54150210189823859099802798523, 5.42510254810447795726185271561, 6.45136232710592368889883233471, 6.95512054653585480974233314849, 8.395287926742891746683400570219, 9.139243844496448643502363350997, 9.968749540253452170107207438533
