| L(s) = 1 | − 0.589·2-s − 1.65·4-s + 4.68·7-s + 2.15·8-s + 1.73·11-s + 0.0918·13-s − 2.76·14-s + 2.03·16-s + 0.766·17-s + 2.05·19-s − 1.02·22-s − 0.205·23-s − 0.0541·26-s − 7.74·28-s − 1.64·29-s + 9.69·31-s − 5.50·32-s − 0.451·34-s − 37-s − 1.21·38-s + 4.33·41-s − 2.82·43-s − 2.87·44-s + 0.121·46-s − 7.39·47-s + 14.9·49-s − 0.151·52-s + ⋯ |
| L(s) = 1 | − 0.417·2-s − 0.826·4-s + 1.77·7-s + 0.761·8-s + 0.524·11-s + 0.0254·13-s − 0.739·14-s + 0.508·16-s + 0.185·17-s + 0.471·19-s − 0.218·22-s − 0.0429·23-s − 0.0106·26-s − 1.46·28-s − 0.306·29-s + 1.74·31-s − 0.973·32-s − 0.0775·34-s − 0.164·37-s − 0.196·38-s + 0.676·41-s − 0.431·43-s − 0.433·44-s + 0.0179·46-s − 1.07·47-s + 2.14·49-s − 0.0210·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976597586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976597586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.589T + 2T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 0.0918T + 13T^{2} \) |
| 17 | \( 1 - 0.766T + 17T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 + 0.205T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 9.69T + 31T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 0.0376T + 59T^{2} \) |
| 61 | \( 1 - 8.97T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 8.10T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 0.973T + 97T^{2} \) |
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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
−7.978917420624267604063846560601, −7.37875929497414366512647869371, −6.51264460684560333230251219805, −5.43117081815804914797906258673, −5.06001955577043427249381584131, −4.30687968799410430714551022265, −3.74726723123611888993913526650, −2.47022112923669588000222897183, −1.47420552522840079493692333347, −0.842701944926613307520907780559,
0.842701944926613307520907780559, 1.47420552522840079493692333347, 2.47022112923669588000222897183, 3.74726723123611888993913526650, 4.30687968799410430714551022265, 5.06001955577043427249381584131, 5.43117081815804914797906258673, 6.51264460684560333230251219805, 7.37875929497414366512647869371, 7.978917420624267604063846560601
