| L(s) = 1 | + 17.4·2-s + 86.8·3-s − 209.·4-s − 1.22e3·5-s + 1.51e3·6-s − 4.75e3·7-s − 1.25e4·8-s − 1.21e4·9-s − 2.12e4·10-s + 2.73e4·11-s − 1.81e4·12-s + 3.84e4·13-s − 8.28e4·14-s − 1.06e5·15-s − 1.11e5·16-s − 1.79e5·17-s − 2.11e5·18-s − 2.48e5·19-s + 2.55e5·20-s − 4.12e5·21-s + 4.75e5·22-s − 2.79e5·23-s − 1.08e6·24-s − 4.60e5·25-s + 6.68e5·26-s − 2.76e6·27-s + 9.94e5·28-s + ⋯ |
| L(s) = 1 | + 0.769·2-s + 0.618·3-s − 0.408·4-s − 0.874·5-s + 0.475·6-s − 0.748·7-s − 1.08·8-s − 0.617·9-s − 0.672·10-s + 0.562·11-s − 0.252·12-s + 0.372·13-s − 0.576·14-s − 0.540·15-s − 0.425·16-s − 0.522·17-s − 0.474·18-s − 0.437·19-s + 0.356·20-s − 0.463·21-s + 0.432·22-s − 0.208·23-s − 0.670·24-s − 0.235·25-s + 0.286·26-s − 1.00·27-s + 0.305·28-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 | \( 1 - 17.4T + 512T^{2} \) |
| 3 | \( 1 - 86.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.22e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.75e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.73e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.84e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.48e5T + 3.22e11T^{2} \) |
| 29 | \( 1 - 1.71e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.16e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.29e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.01e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.57e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.30e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.45e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.87e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.74e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.11e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.39e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.38e6T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.96e7T + 7.60e17T^{2} \) |
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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
−14.87430093519048577658362422274, −13.87112277759951134772736484034, −12.73074504344977351826398388462, −11.48192949061882643660771020084, −9.383301460647405449220685352568, −8.240106012873132503020332058887, −6.22112520851791918325394494829, −4.23130238076880100337200484341, −3.09118033530364278605688855992, 0,
3.09118033530364278605688855992, 4.23130238076880100337200484341, 6.22112520851791918325394494829, 8.240106012873132503020332058887, 9.383301460647405449220685352568, 11.48192949061882643660771020084, 12.73074504344977351826398388462, 13.87112277759951134772736484034, 14.87430093519048577658362422274
