| L(s) = 1 | + 473.·2-s + 2.40e3·3-s + 9.35e4·4-s − 3.13e5·5-s + 1.13e6·6-s + 1.02e7·7-s − 1.77e7·8-s − 1.23e8·9-s − 1.48e8·10-s + 2.58e8·11-s + 2.24e8·12-s − 2.56e9·13-s + 4.85e9·14-s − 7.51e8·15-s − 2.06e10·16-s − 3.42e9·17-s − 5.84e10·18-s − 3.14e9·19-s − 2.92e10·20-s + 2.45e10·21-s + 1.22e11·22-s − 7.83e10·23-s − 4.27e10·24-s − 6.64e11·25-s − 1.21e12·26-s − 6.06e11·27-s + 9.57e11·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.211·3-s + 0.713·4-s − 0.358·5-s + 0.276·6-s + 0.671·7-s − 0.374·8-s − 0.955·9-s − 0.469·10-s + 0.363·11-s + 0.150·12-s − 0.873·13-s + 0.878·14-s − 0.0757·15-s − 1.20·16-s − 0.118·17-s − 1.25·18-s − 0.0424·19-s − 0.255·20-s + 0.141·21-s + 0.476·22-s − 0.208·23-s − 0.0792·24-s − 0.871·25-s − 1.14·26-s − 0.413·27-s + 0.479·28-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 | \( 1 - 473.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 2.40e3T + 1.29e8T^{2} \) |
| 5 | \( 1 + 3.13e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.02e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 2.58e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.56e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.42e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.14e9T + 5.48e21T^{2} \) |
| 29 | \( 1 + 3.25e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.84e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.54e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.34e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 2.13e12T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.37e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.66e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.41e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.79e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.38e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 6.35e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.21e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.58e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.25e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.04e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.14e16T + 5.95e33T^{2} \) |
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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
−13.45313944846794527040310167231, −12.08703987034837478697999587904, −11.27410369126127634042316118367, −9.208354841756174454405998322240, −7.72171049684873045620070030195, −5.98293381207186817142794589267, −4.79512128661268637229922583090, −3.57767054953846969560307646013, −2.22062630089728786182807260819, 0,
2.22062630089728786182807260819, 3.57767054953846969560307646013, 4.79512128661268637229922583090, 5.98293381207186817142794589267, 7.72171049684873045620070030195, 9.208354841756174454405998322240, 11.27410369126127634042316118367, 12.08703987034837478697999587904, 13.45313944846794527040310167231
