| L(s) = 1 | + 3.53·2-s − 10.7·3-s − 19.4·4-s − 25·5-s − 38.0·6-s − 227.·7-s − 182.·8-s − 127.·9-s − 88.4·10-s − 121·11-s + 209.·12-s + 680.·13-s − 805.·14-s + 269.·15-s − 21.2·16-s + 547.·17-s − 450.·18-s − 361·19-s + 486.·20-s + 2.44e3·21-s − 428.·22-s − 1.90e3·23-s + 1.96e3·24-s + 625·25-s + 2.40e3·26-s + 3.98e3·27-s + 4.43e3·28-s + ⋯ |
| L(s) = 1 | + 0.625·2-s − 0.690·3-s − 0.608·4-s − 0.447·5-s − 0.431·6-s − 1.75·7-s − 1.00·8-s − 0.523·9-s − 0.279·10-s − 0.301·11-s + 0.420·12-s + 1.11·13-s − 1.09·14-s + 0.308·15-s − 0.0207·16-s + 0.459·17-s − 0.327·18-s − 0.229·19-s + 0.272·20-s + 1.21·21-s − 0.188·22-s − 0.750·23-s + 0.694·24-s + 0.200·25-s + 0.698·26-s + 1.05·27-s + 1.06·28-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 | \( 1 - 3.53T + 32T^{2} \) |
| 3 | \( 1 + 10.7T + 243T^{2} \) |
| 7 | \( 1 + 227.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 680.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 547.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 158.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.67e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.99e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.12e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.79e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.78e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.18e4T + 8.58e9T^{2} \) |
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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
−8.827566296516099427192518248411, −8.031890204420600667816955574965, −6.65563548509785259501429286784, −6.08022645715183488732396513339, −5.49203710722015170945817828752, −4.31945246947356357740076361794, −3.50238811517882268716761865212, −2.83781699082540215201508620973, −0.75275270506533985103058763069, 0,
0.75275270506533985103058763069, 2.83781699082540215201508620973, 3.50238811517882268716761865212, 4.31945246947356357740076361794, 5.49203710722015170945817828752, 6.08022645715183488732396513339, 6.65563548509785259501429286784, 8.031890204420600667816955574965, 8.827566296516099427192518248411
