| L(s) = 1 | − 1.83·2-s + 0.352·3-s + 1.37·4-s − 0.647·6-s + 3.54·7-s + 1.14·8-s − 2.87·9-s − 1.74·11-s + 0.484·12-s − 1.99·13-s − 6.51·14-s − 4.85·16-s − 6.52·17-s + 5.28·18-s − 4.61·19-s + 1.25·21-s + 3.19·22-s + 9.26·23-s + 0.404·24-s + 3.66·26-s − 2.07·27-s + 4.87·28-s − 9.40·29-s + 7.06·31-s + 6.63·32-s − 0.613·33-s + 11.9·34-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.203·3-s + 0.687·4-s − 0.264·6-s + 1.34·7-s + 0.406·8-s − 0.958·9-s − 0.525·11-s + 0.139·12-s − 0.552·13-s − 1.74·14-s − 1.21·16-s − 1.58·17-s + 1.24·18-s − 1.05·19-s + 0.272·21-s + 0.682·22-s + 1.93·23-s + 0.0826·24-s + 0.717·26-s − 0.398·27-s + 0.921·28-s − 1.74·29-s + 1.26·31-s + 1.17·32-s − 0.106·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 - 0.352T + 3T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 - 9.26T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 8.12T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 7.83T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−9.345138322691051029266095528530, −8.806038797921591573379824531838, −8.179498431412592056800751489835, −7.51182727520342744551379314911, −6.52801152651150597522371190630, −5.11826010551149792835613094366, −4.48116997824360617345329077200, −2.67337699010519136620551459981, −1.72231313287446702179329592662, 0,
1.72231313287446702179329592662, 2.67337699010519136620551459981, 4.48116997824360617345329077200, 5.11826010551149792835613094366, 6.52801152651150597522371190630, 7.51182727520342744551379314911, 8.179498431412592056800751489835, 8.806038797921591573379824531838, 9.345138322691051029266095528530
