| L(s) = 1 | − 10.7·2-s − 11.3·3-s + 83.8·4-s + 25·5-s + 122.·6-s − 59.3·7-s − 557.·8-s − 114.·9-s − 269.·10-s + 121·11-s − 951.·12-s + 781.·13-s + 638.·14-s − 283.·15-s + 3.32e3·16-s + 774.·17-s + 1.22e3·18-s + 361·19-s + 2.09e3·20-s + 673.·21-s − 1.30e3·22-s + 2.90e3·23-s + 6.33e3·24-s + 625·25-s − 8.40e3·26-s + 4.05e3·27-s − 4.97e3·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.728·3-s + 2.61·4-s + 0.447·5-s + 1.38·6-s − 0.457·7-s − 3.08·8-s − 0.469·9-s − 0.850·10-s + 0.301·11-s − 1.90·12-s + 1.28·13-s + 0.870·14-s − 0.325·15-s + 3.24·16-s + 0.649·17-s + 0.893·18-s + 0.229·19-s + 1.17·20-s + 0.333·21-s − 0.573·22-s + 1.14·23-s + 2.24·24-s + 0.200·25-s − 2.43·26-s + 1.07·27-s − 1.19·28-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
| \(L(3)\) |
\(\approx\) |
\(0.8232984147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8232984147\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 10.7T + 32T^{2} \) |
| 3 | \( 1 + 11.3T + 243T^{2} \) |
| 7 | \( 1 + 59.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 781.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 774.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.98e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.57e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.61e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.95e4T + 8.58e9T^{2} \) |
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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
−9.148921739100589671797925420576, −8.620320946856510241019492385534, −7.67849645983905555999799911095, −6.71184579985472273299159780878, −6.15660871716597498550084230537, −5.40771739888719749194970654666, −3.48471289738462765370111272511, −2.48469276119001125621601300595, −1.20035607090913465651494966823, −0.64617400699796922027558597647,
0.64617400699796922027558597647, 1.20035607090913465651494966823, 2.48469276119001125621601300595, 3.48471289738462765370111272511, 5.40771739888719749194970654666, 6.15660871716597498550084230537, 6.71184579985472273299159780878, 7.67849645983905555999799911095, 8.620320946856510241019492385534, 9.148921739100589671797925420576
