| L(s) = 1 | + 113.·2-s − 1.73e4·3-s − 1.18e5·4-s + 1.00e5·5-s − 1.97e6·6-s + 7.81e6·7-s − 2.83e7·8-s + 1.71e8·9-s + 1.14e7·10-s − 3.42e8·11-s + 2.04e9·12-s + 2.40e9·13-s + 8.89e8·14-s − 1.73e9·15-s + 1.22e10·16-s + 3.54e10·17-s + 1.95e10·18-s + 7.34e10·19-s − 1.18e10·20-s − 1.35e11·21-s − 3.89e10·22-s − 7.83e10·23-s + 4.91e11·24-s − 7.52e11·25-s + 2.74e11·26-s − 7.33e11·27-s − 9.23e11·28-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 1.52·3-s − 0.901·4-s + 0.114·5-s − 0.479·6-s + 0.512·7-s − 0.597·8-s + 1.32·9-s + 0.0360·10-s − 0.481·11-s + 1.37·12-s + 0.818·13-s + 0.161·14-s − 0.175·15-s + 0.713·16-s + 1.23·17-s + 0.417·18-s + 0.992·19-s − 0.103·20-s − 0.781·21-s − 0.151·22-s − 0.208·23-s + 0.911·24-s − 0.986·25-s + 0.257·26-s − 0.499·27-s − 0.461·28-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 | \( 1 - 113.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.73e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.00e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 7.81e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.42e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.40e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.54e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.34e10T + 5.48e21T^{2} \) |
| 29 | \( 1 + 3.80e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 6.18e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.58e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.70e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 3.28e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.82e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 4.78e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.17e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.03e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.88e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 6.67e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.33e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.47e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 6.08e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.87e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.37e16T + 5.95e33T^{2} \) |
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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
−13.10518854764255112493522419207, −11.99140918878363046211663615339, −10.89105960371875666694624321155, −9.550218808585970952825993186410, −7.75191902965920229473551400074, −5.79985039285451090459054665518, −5.25533057115416513100635401454, −3.77836108506994807363868687696, −1.21816553566593693198062868756, 0,
1.21816553566593693198062868756, 3.77836108506994807363868687696, 5.25533057115416513100635401454, 5.79985039285451090459054665518, 7.75191902965920229473551400074, 9.550218808585970952825993186410, 10.89105960371875666694624321155, 11.99140918878363046211663615339, 13.10518854764255112493522419207
