| L(s) = 1 | + 63.0·2-s − 462.·3-s + 1.92e3·4-s + 5.90e3·5-s − 2.91e4·6-s − 3.66e4·7-s − 7.62e3·8-s + 3.63e4·9-s + 3.72e5·10-s + 1.96e4·11-s − 8.90e5·12-s − 1.25e6·13-s − 2.31e6·14-s − 2.72e6·15-s − 4.42e6·16-s − 6.08e6·17-s + 2.29e6·18-s − 6.45e6·19-s + 1.13e7·20-s + 1.69e7·21-s + 1.24e6·22-s + 6.43e6·23-s + 3.52e6·24-s − 1.39e7·25-s − 7.93e7·26-s + 6.50e7·27-s − 7.07e7·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.09·3-s + 0.940·4-s + 0.845·5-s − 1.52·6-s − 0.825·7-s − 0.0822·8-s + 0.205·9-s + 1.17·10-s + 0.0368·11-s − 1.03·12-s − 0.940·13-s − 1.14·14-s − 0.928·15-s − 1.05·16-s − 1.04·17-s + 0.285·18-s − 0.597·19-s + 0.795·20-s + 0.905·21-s + 0.0513·22-s + 0.208·23-s + 0.0903·24-s − 0.285·25-s − 1.31·26-s + 0.872·27-s − 0.776·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 6.43e6T \) |
| good | 2 | \( 1 - 63.0T + 2.04e3T^{2} \) |
| 3 | \( 1 + 462.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 5.90e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.66e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.96e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.25e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.08e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 6.45e6T + 1.16e14T^{2} \) |
| 29 | \( 1 - 4.85e6T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.74e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.61e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.61e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.63e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.26e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.77e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.62e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.02e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.86e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 5.53e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.09e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.09e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.23e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 5.03e10T + 7.15e21T^{2} \) |
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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
−14.36335859927652560808250098699, −13.16164201778262473170534430034, −12.33401589371837773080764328575, −11.03451850723066388128100894862, −9.474591655123010788672120406097, −6.58703709750858364622757275931, −5.79919448008257567738931099685, −4.54120126063288495215467393208, −2.58213292769192526177987910309, 0,
2.58213292769192526177987910309, 4.54120126063288495215467393208, 5.79919448008257567738931099685, 6.58703709750858364622757275931, 9.474591655123010788672120406097, 11.03451850723066388128100894862, 12.33401589371837773080764328575, 13.16164201778262473170534430034, 14.36335859927652560808250098699
