| L(s) = 1 | + 1.46·3-s − 5-s − 1.53·7-s − 0.860·9-s − 0.860·11-s − 0.139·13-s − 1.46·15-s + 5.50·17-s + 5.25·19-s − 2.24·21-s + 23-s + 25-s − 5.64·27-s − 9.76·29-s − 6.78·31-s − 1.25·33-s + 1.53·35-s + 12.0·37-s − 0.203·39-s − 9.98·41-s + 11.4·43-s + 0.860·45-s + 2.32·47-s − 4.63·49-s + 8.05·51-s − 0.149·53-s + 0.860·55-s + ⋯ |
| L(s) = 1 | + 0.844·3-s − 0.447·5-s − 0.581·7-s − 0.286·9-s − 0.259·11-s − 0.0386·13-s − 0.377·15-s + 1.33·17-s + 1.20·19-s − 0.490·21-s + 0.208·23-s + 0.200·25-s − 1.08·27-s − 1.81·29-s − 1.21·31-s − 0.219·33-s + 0.259·35-s + 1.98·37-s − 0.0325·39-s − 1.55·41-s + 1.74·43-s + 0.128·45-s + 0.338·47-s − 0.662·49-s + 1.12·51-s − 0.0205·53-s + 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 0.860T + 11T^{2} \) |
| 13 | \( 1 + 0.139T + 13T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 + 6.78T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + 0.149T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 7.11T + 73T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−7.54875307819242281613166725110, −7.27957318184704056133901414442, −5.93504393991789923372366009272, −5.65087954329435709629557833349, −4.62762429233292709234485947794, −3.53360930879899913626932836946, −3.32369854638856356581758360762, −2.47575935989299656273026977826, −1.33311487616825555007860774848, 0,
1.33311487616825555007860774848, 2.47575935989299656273026977826, 3.32369854638856356581758360762, 3.53360930879899913626932836946, 4.62762429233292709234485947794, 5.65087954329435709629557833349, 5.93504393991789923372366009272, 7.27957318184704056133901414442, 7.54875307819242281613166725110
