| L(s) = 1 | − 519.·2-s + 1.33e4·3-s + 1.38e5·4-s + 3.29e5·5-s − 6.94e6·6-s + 1.47e7·7-s − 3.82e6·8-s + 4.98e7·9-s − 1.71e8·10-s − 3.09e8·11-s + 1.85e9·12-s − 5.57e9·13-s − 7.67e9·14-s + 4.41e9·15-s − 1.61e10·16-s − 4.02e10·17-s − 2.58e10·18-s + 1.16e11·19-s + 4.56e10·20-s + 1.97e11·21-s + 1.60e11·22-s − 7.83e10·23-s − 5.11e10·24-s − 6.54e11·25-s + 2.89e12·26-s − 1.06e12·27-s + 2.04e12·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.17·3-s + 1.05·4-s + 0.377·5-s − 1.68·6-s + 0.968·7-s − 0.0805·8-s + 0.386·9-s − 0.541·10-s − 0.435·11-s + 1.24·12-s − 1.89·13-s − 1.38·14-s + 0.444·15-s − 0.940·16-s − 1.39·17-s − 0.553·18-s + 1.57·19-s + 0.398·20-s + 1.14·21-s + 0.623·22-s − 0.208·23-s − 0.0948·24-s − 0.857·25-s + 2.71·26-s − 0.722·27-s + 1.02·28-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 | \( 1 + 519.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.33e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 3.29e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.47e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.09e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 5.57e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.02e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.16e11T + 5.48e21T^{2} \) |
| 29 | \( 1 - 2.70e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 4.10e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.24e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.25e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 5.34e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 3.17e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.36e13T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.43e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.52e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 5.39e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 5.08e14T + 2.96e31T^{2} \) |
| 73 | \( 1 + 5.69e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.08e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 9.04e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.54e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 5.46e16T + 5.95e33T^{2} \) |
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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
−13.59583915798910943443841206119, −11.60985418079714403294319049224, −10.09077893867368471643238705390, −9.213155476516617900532248892675, −8.116239258466034329967661599612, −7.30973708167963156393989380473, −4.84279905721200289171300547048, −2.59783455981676816998718721926, −1.73227373038136342241379649448, 0,
1.73227373038136342241379649448, 2.59783455981676816998718721926, 4.84279905721200289171300547048, 7.30973708167963156393989380473, 8.116239258466034329967661599612, 9.213155476516617900532248892675, 10.09077893867368471643238705390, 11.60985418079714403294319049224, 13.59583915798910943443841206119
