| L(s) = 1 | + 638.·2-s − 7.19e3·3-s + 2.76e5·4-s + 6.03e5·5-s − 4.58e6·6-s − 2.24e7·7-s + 9.25e7·8-s − 7.73e7·9-s + 3.84e8·10-s − 7.48e8·11-s − 1.98e9·12-s − 1.54e9·13-s − 1.43e10·14-s − 4.33e9·15-s + 2.28e10·16-s − 1.78e10·17-s − 4.93e10·18-s + 1.25e11·19-s + 1.66e11·20-s + 1.61e11·21-s − 4.77e11·22-s − 7.83e10·23-s − 6.65e11·24-s − 3.98e11·25-s − 9.86e11·26-s + 1.48e12·27-s − 6.19e12·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.632·3-s + 2.10·4-s + 0.690·5-s − 1.11·6-s − 1.47·7-s + 1.94·8-s − 0.599·9-s + 1.21·10-s − 1.05·11-s − 1.33·12-s − 0.525·13-s − 2.59·14-s − 0.437·15-s + 1.32·16-s − 0.621·17-s − 1.05·18-s + 1.69·19-s + 1.45·20-s + 0.930·21-s − 1.85·22-s − 0.208·23-s − 1.23·24-s − 0.522·25-s − 0.926·26-s + 1.01·27-s − 3.09·28-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 | \( 1 - 638.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 7.19e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 6.03e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 2.24e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.48e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.54e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.78e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.25e11T + 5.48e21T^{2} \) |
| 29 | \( 1 + 9.63e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 6.60e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.57e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 7.52e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 2.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 7.41e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.95e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 4.75e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.57e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.44e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 5.55e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.80e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.03e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 8.33e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 5.37e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.35e17T + 5.95e33T^{2} \) |
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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
−13.23486571921947270106296818623, −12.40372397069522708975583903497, −11.13955781888690950036544266924, −9.722848683894748147160464314890, −7.05847838755477040844576923181, −5.85764532911800489509082464224, −5.24529550261364261168299467191, −3.41196285450496793410421012733, −2.39417189281797810053116970563, 0,
2.39417189281797810053116970563, 3.41196285450496793410421012733, 5.24529550261364261168299467191, 5.85764532911800489509082464224, 7.05847838755477040844576923181, 9.722848683894748147160464314890, 11.13955781888690950036544266924, 12.40372397069522708975583903497, 13.23486571921947270106296818623
