| L(s) = 1 | − 6.95·2-s − 10.9·3-s + 16.4·4-s + 25·5-s + 76.0·6-s − 81.2·7-s + 108.·8-s − 123.·9-s − 173.·10-s − 121·11-s − 179.·12-s − 773.·13-s + 565.·14-s − 273.·15-s − 1.27e3·16-s − 1.24e3·17-s + 858.·18-s + 361·19-s + 410.·20-s + 888.·21-s + 841.·22-s + 4.38e3·23-s − 1.18e3·24-s + 625·25-s + 5.38e3·26-s + 4.00e3·27-s − 1.33e3·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.701·3-s + 0.513·4-s + 0.447·5-s + 0.862·6-s − 0.626·7-s + 0.598·8-s − 0.508·9-s − 0.550·10-s − 0.301·11-s − 0.359·12-s − 1.26·13-s + 0.770·14-s − 0.313·15-s − 1.24·16-s − 1.04·17-s + 0.624·18-s + 0.229·19-s + 0.229·20-s + 0.439·21-s + 0.370·22-s + 1.72·23-s − 0.420·24-s + 0.200·25-s + 1.56·26-s + 1.05·27-s − 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 6.95T + 32T^{2} \) |
| 3 | \( 1 + 10.9T + 243T^{2} \) |
| 7 | \( 1 + 81.2T + 1.68e4T^{2} \) |
| 13 | \( 1 + 773.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.24e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.88e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.86e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.70e4T + 8.58e9T^{2} \) |
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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
−8.943419510026556256698418487668, −8.147099958751080640753209390473, −6.96566318450561330138217817115, −6.63573699686845762315370244659, −5.27042544172047029762484680248, −4.76637240838442584626742867988, −3.06588545419133266063261157323, −2.08908887282167593891363733331, −0.75781940868704004289334124887, 0,
0.75781940868704004289334124887, 2.08908887282167593891363733331, 3.06588545419133266063261157323, 4.76637240838442584626742867988, 5.27042544172047029762484680248, 6.63573699686845762315370244659, 6.96566318450561330138217817115, 8.147099958751080640753209390473, 8.943419510026556256698418487668
