| L(s) = 1 | − 30.1·2-s − 193.·3-s + 396.·4-s + 568.·5-s + 5.82e3·6-s + 3.61e3·7-s + 3.47e3·8-s + 1.77e4·9-s − 1.71e4·10-s − 2.70e3·11-s − 7.67e4·12-s + 2.28e4·13-s − 1.08e5·14-s − 1.10e5·15-s − 3.07e5·16-s − 9.38e4·17-s − 5.33e5·18-s + 2.69e5·19-s + 2.25e5·20-s − 6.98e5·21-s + 8.16e4·22-s − 2.79e5·23-s − 6.71e5·24-s − 1.62e6·25-s − 6.87e5·26-s + 3.81e5·27-s + 1.43e6·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 1.37·3-s + 0.774·4-s + 0.407·5-s + 1.83·6-s + 0.568·7-s + 0.299·8-s + 0.899·9-s − 0.542·10-s − 0.0558·11-s − 1.06·12-s + 0.221·13-s − 0.758·14-s − 0.561·15-s − 1.17·16-s − 0.272·17-s − 1.19·18-s + 0.473·19-s + 0.315·20-s − 0.784·21-s + 0.0743·22-s − 0.208·23-s − 0.413·24-s − 0.834·25-s − 0.295·26-s + 0.138·27-s + 0.440·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 2.79e5T \) |
| good | 2 | \( 1 + 30.1T + 512T^{2} \) |
| 3 | \( 1 + 193.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 568.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.61e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.70e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.28e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.38e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.69e5T + 3.22e11T^{2} \) |
| 29 | \( 1 - 3.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.81e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.47e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.70e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.27e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.16e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.29e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.75e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.95e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.76e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.66e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.67e8T + 7.60e17T^{2} \) |
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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
−15.77708992971716726957703690553, −13.73557839846143718904579728132, −11.90774542554814680314225072875, −10.87620299379540148338950967648, −9.840583207675360207850011722128, −8.239369191638625955044711583831, −6.60056005849286683270383785777, −4.99912128300218473283783112113, −1.47351254318745574544285578858, 0,
1.47351254318745574544285578858, 4.99912128300218473283783112113, 6.60056005849286683270383785777, 8.239369191638625955044711583831, 9.840583207675360207850011722128, 10.87620299379540148338950967648, 11.90774542554814680314225072875, 13.73557839846143718904579728132, 15.77708992971716726957703690553
