| L(s) = 1 | + 4.16·2-s − 15.7·3-s − 14.6·4-s − 25·5-s − 65.7·6-s − 177.·7-s − 194.·8-s + 6.42·9-s − 104.·10-s + 121·11-s + 231.·12-s − 531.·13-s − 737.·14-s + 394.·15-s − 341.·16-s + 1.44e3·17-s + 26.7·18-s + 361·19-s + 366.·20-s + 2.79e3·21-s + 504.·22-s − 2.14e3·23-s + 3.06e3·24-s + 625·25-s − 2.21e3·26-s + 3.73e3·27-s + 2.59e3·28-s + ⋯ |
| L(s) = 1 | + 0.736·2-s − 1.01·3-s − 0.457·4-s − 0.447·5-s − 0.746·6-s − 1.36·7-s − 1.07·8-s + 0.0264·9-s − 0.329·10-s + 0.301·11-s + 0.463·12-s − 0.872·13-s − 1.00·14-s + 0.453·15-s − 0.333·16-s + 1.21·17-s + 0.0194·18-s + 0.229·19-s + 0.204·20-s + 1.38·21-s + 0.222·22-s − 0.846·23-s + 1.08·24-s + 0.200·25-s − 0.642·26-s + 0.986·27-s + 0.624·28-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 | \( 1 - 4.16T + 32T^{2} \) |
| 3 | \( 1 + 15.7T + 243T^{2} \) |
| 7 | \( 1 + 177.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 531.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 564.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 950.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.74e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.28e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.88e4T + 8.58e9T^{2} \) |
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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
−8.917893306776499316179926105765, −7.79757589595198917040457458599, −6.76735208529339583511528798309, −5.97216707311696320661720378808, −5.43700268482721085413544010270, −4.45059097976514859492384345442, −3.56692175079758058199295927837, −2.76676459080675261690632314931, −0.75121472734852799101567191390, 0,
0.75121472734852799101567191390, 2.76676459080675261690632314931, 3.56692175079758058199295927837, 4.45059097976514859492384345442, 5.43700268482721085413544010270, 5.97216707311696320661720378808, 6.76735208529339583511528798309, 7.79757589595198917040457458599, 8.917893306776499316179926105765
