L(s) = 1 | − 174.·2-s − 1.54e4·3-s − 1.00e5·4-s − 1.53e6·5-s + 2.68e6·6-s − 1.10e7·7-s + 4.03e7·8-s + 1.09e8·9-s + 2.67e8·10-s + 7.06e8·11-s + 1.55e9·12-s − 1.64e9·13-s + 1.91e9·14-s + 2.37e10·15-s + 6.16e9·16-s − 9.14e9·17-s − 1.90e10·18-s − 3.02e10·19-s + 1.54e11·20-s + 1.69e11·21-s − 1.23e11·22-s − 7.83e10·23-s − 6.23e11·24-s + 1.60e12·25-s + 2.85e11·26-s + 3.08e11·27-s + 1.10e12·28-s + ⋯ |
L(s) = 1 | − 0.481·2-s − 1.35·3-s − 0.768·4-s − 1.76·5-s + 0.653·6-s − 0.721·7-s + 0.851·8-s + 0.845·9-s + 0.847·10-s + 0.994·11-s + 1.04·12-s − 0.557·13-s + 0.347·14-s + 2.39·15-s + 0.358·16-s − 0.317·17-s − 0.406·18-s − 0.408·19-s + 1.35·20-s + 0.979·21-s − 0.478·22-s − 0.208·23-s − 1.15·24-s + 2.10·25-s + 0.268·26-s + 0.210·27-s + 0.554·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + 7.83e10T \) |
good | 2 | \( 1 + 174.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.54e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 1.53e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.10e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 7.06e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.64e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 9.14e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.02e10T + 5.48e21T^{2} \) |
| 29 | \( 1 + 1.73e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 4.09e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.45e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 4.13e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 4.40e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 3.15e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.38e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.65e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.88e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 3.94e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.31e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.50e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.52e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.69e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 7.01e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 6.70e16T + 5.95e33T^{2} \) |
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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
−12.79628558715962558783180576948, −11.85412317379327265898182405950, −10.86594465643882765570182246115, −9.347780324244358978459139862806, −7.86564039859984670580087028581, −6.53525704898095400395476414564, −4.75960401657025647632905403876, −3.78049635718686893592436336604, −0.75607542427628276910396769241, 0,
0.75607542427628276910396769241, 3.78049635718686893592436336604, 4.75960401657025647632905403876, 6.53525704898095400395476414564, 7.86564039859984670580087028581, 9.347780324244358978459139862806, 10.86594465643882765570182246115, 11.85412317379327265898182405950, 12.79628558715962558783180576948