| L(s) = 1 | + 3·2-s − 3.72·3-s + 4-s − 11.1·6-s + 25.6·7-s − 21·8-s − 13.1·9-s + 12.6·11-s − 3.72·12-s − 8.16·13-s + 76.8·14-s − 71·16-s + 76.0·17-s − 39.4·18-s − 103.·19-s − 95.2·21-s + 37.8·22-s + 23·23-s + 78.1·24-s − 24.4·26-s + 149.·27-s + 25.6·28-s − 267.·29-s − 63.7·31-s − 45·32-s − 46.8·33-s + 228.·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.715·3-s + 0.125·4-s − 0.759·6-s + 1.38·7-s − 0.928·8-s − 0.487·9-s + 0.345·11-s − 0.0894·12-s − 0.174·13-s + 1.46·14-s − 1.10·16-s + 1.08·17-s − 0.516·18-s − 1.24·19-s − 0.989·21-s + 0.366·22-s + 0.208·23-s + 0.664·24-s − 0.184·26-s + 1.06·27-s + 0.172·28-s − 1.71·29-s − 0.369·31-s − 0.248·32-s − 0.247·33-s + 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3T + 8T^{2} \) |
| 3 | \( 1 + 3.72T + 27T^{2} \) |
| 7 | \( 1 - 25.6T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.16T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 63.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 239.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 2.31T + 1.48e5T^{2} \) |
| 59 | \( 1 - 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 294.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 286.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 551.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 21.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
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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
−10.10173519987673804045235701880, −8.851912615893465499607307107092, −8.147521030609398738643487918778, −6.87322720082784842523604662897, −5.78419010741016477623843507553, −5.22436733394597780853650167262, −4.42210390595879919469799024763, −3.30311822301414627958047823141, −1.74695349749185104224762682292, 0,
1.74695349749185104224762682292, 3.30311822301414627958047823141, 4.42210390595879919469799024763, 5.22436733394597780853650167262, 5.78419010741016477623843507553, 6.87322720082784842523604662897, 8.147521030609398738643487918778, 8.851912615893465499607307107092, 10.10173519987673804045235701880
