L(s) = 1 | − 69.2·2-s − 806.·3-s + 2.74e3·4-s − 1.05e4·5-s + 5.58e4·6-s − 3.92e4·7-s − 4.82e4·8-s + 4.72e5·9-s + 7.31e5·10-s + 5.62e5·11-s − 2.21e6·12-s − 7.14e5·13-s + 2.71e6·14-s + 8.51e6·15-s − 2.27e6·16-s − 4.57e6·17-s − 3.27e7·18-s + 4.10e6·19-s − 2.90e7·20-s + 3.16e7·21-s − 3.89e7·22-s + 6.43e6·23-s + 3.89e7·24-s + 6.28e7·25-s + 4.94e7·26-s − 2.38e8·27-s − 1.07e8·28-s + ⋯ |
L(s) = 1 | − 1.52·2-s − 1.91·3-s + 1.34·4-s − 1.51·5-s + 2.93·6-s − 0.882·7-s − 0.521·8-s + 2.66·9-s + 2.31·10-s + 1.05·11-s − 2.56·12-s − 0.533·13-s + 1.35·14-s + 2.89·15-s − 0.543·16-s − 0.782·17-s − 4.08·18-s + 0.380·19-s − 2.02·20-s + 1.69·21-s − 1.61·22-s + 0.208·23-s + 0.998·24-s + 1.28·25-s + 0.816·26-s − 3.19·27-s − 1.18·28-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 | \( 1 + 69.2T + 2.04e3T^{2} \) |
| 3 | \( 1 + 806.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.05e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.92e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.62e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 7.14e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.57e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.10e6T + 1.16e14T^{2} \) |
| 29 | \( 1 - 1.74e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.09e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.14e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.92e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.99e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.05e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.05e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.13e7T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.65e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.15e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 5.03e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.30e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.77e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 8.51e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.24e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.95e10T + 7.15e21T^{2} \) |
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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
−15.62289246503140552585928969759, −12.46542599954661207217839977698, −11.57655336308808370151234035005, −10.70753770721537750544560609360, −9.350897528248305854084951886462, −7.42917209220168957904522653755, −6.52967549761884850141539522426, −4.33139547589029234116406760361, −0.880698169612953561047549875155, 0,
0.880698169612953561047549875155, 4.33139547589029234116406760361, 6.52967549761884850141539522426, 7.42917209220168957904522653755, 9.350897528248305854084951886462, 10.70753770721537750544560609360, 11.57655336308808370151234035005, 12.46542599954661207217839977698, 15.62289246503140552585928969759