| L(s) = 1 | − 0.618·3-s − 2.85·5-s − 1.23·7-s − 2.61·9-s + 3.61·11-s + 3.85·13-s + 1.76·15-s + 4.47·17-s + 4.47·19-s + 0.763·21-s + 3.85·23-s + 3.14·25-s + 3.47·27-s + 6.32·29-s − 9.61·31-s − 2.23·33-s + 3.52·35-s − 37-s − 2.38·39-s + 7.38·41-s + 0.763·43-s + 7.47·45-s − 3.23·47-s − 5.47·49-s − 2.76·51-s − 8.47·53-s − 10.3·55-s + ⋯ |
| L(s) = 1 | − 0.356·3-s − 1.27·5-s − 0.467·7-s − 0.872·9-s + 1.09·11-s + 1.06·13-s + 0.455·15-s + 1.08·17-s + 1.02·19-s + 0.166·21-s + 0.803·23-s + 0.629·25-s + 0.668·27-s + 1.17·29-s − 1.72·31-s − 0.389·33-s + 0.596·35-s − 0.164·37-s − 0.381·39-s + 1.15·41-s + 0.116·43-s + 1.11·45-s − 0.472·47-s − 0.781·49-s − 0.387·51-s − 1.16·53-s − 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.002028253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002028253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 8.38T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−11.09848767935619698885534938038, −9.758196078773160442988747826267, −8.851162501378949552196141666241, −8.073159967741036873319422779576, −7.09830292767490459361795472967, −6.17719615950387939808892690541, −5.18225365012140719876428920572, −3.79428516427035269298902039944, −3.23448319847425179167872990399, −0.922288737221914523837860158115,
0.922288737221914523837860158115, 3.23448319847425179167872990399, 3.79428516427035269298902039944, 5.18225365012140719876428920572, 6.17719615950387939808892690541, 7.09830292767490459361795472967, 8.073159967741036873319422779576, 8.851162501378949552196141666241, 9.758196078773160442988747826267, 11.09848767935619698885534938038
