| L(s) = 1 | + 4.96·2-s − 9.59·3-s + 16.6·4-s − 5·5-s − 47.6·6-s − 3.47·7-s + 42.8·8-s + 65.0·9-s − 24.8·10-s − 11·11-s − 159.·12-s + 11.2·13-s − 17.2·14-s + 47.9·15-s + 79.4·16-s − 2.14·17-s + 322.·18-s + 19·19-s − 83.1·20-s + 33.3·21-s − 54.5·22-s + 117.·23-s − 410.·24-s + 25·25-s + 55.7·26-s − 365.·27-s − 57.7·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 1.84·3-s + 2.07·4-s − 0.447·5-s − 3.23·6-s − 0.187·7-s + 1.89·8-s + 2.40·9-s − 0.784·10-s − 0.301·11-s − 3.83·12-s + 0.239·13-s − 0.328·14-s + 0.825·15-s + 1.24·16-s − 0.0306·17-s + 4.22·18-s + 0.229·19-s − 0.929·20-s + 0.346·21-s − 0.529·22-s + 1.06·23-s − 3.49·24-s + 0.200·25-s + 0.420·26-s − 2.60·27-s − 0.389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 4.96T + 8T^{2} \) |
| 3 | \( 1 + 9.59T + 27T^{2} \) |
| 7 | \( 1 + 3.47T + 343T^{2} \) |
| 13 | \( 1 - 11.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.14T + 4.91e3T^{2} \) |
| 23 | \( 1 - 117.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 431.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 39.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 44.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 770.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 843.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 475.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 605.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 118.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 550.T + 9.12e5T^{2} \) |
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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
−9.427601720884995921375714944265, −7.70050525614920174916614129758, −6.92318368981481894045947625767, −6.26299067588290300039334897880, −5.50542190799375244070698141992, −4.89234144636434192099953867418, −4.14211835857289026086553504781, −3.11959324979876191163899397296, −1.49666904229627869351072453029, 0,
1.49666904229627869351072453029, 3.11959324979876191163899397296, 4.14211835857289026086553504781, 4.89234144636434192099953867418, 5.50542190799375244070698141992, 6.26299067588290300039334897880, 6.92318368981481894045947625767, 7.70050525614920174916614129758, 9.427601720884995921375714944265
