| L(s) = 1 | + 8.98·2-s − 20.3·3-s + 48.7·4-s + 25·5-s − 182.·6-s + 110.·7-s + 150.·8-s + 170.·9-s + 224.·10-s − 121·11-s − 992.·12-s + 216.·13-s + 996.·14-s − 508.·15-s − 204.·16-s − 40.2·17-s + 1.53e3·18-s − 361·19-s + 1.21e3·20-s − 2.25e3·21-s − 1.08e3·22-s + 1.74e3·23-s − 3.06e3·24-s + 625·25-s + 1.94e3·26-s + 1.47e3·27-s + 5.41e3·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 1.30·3-s + 1.52·4-s + 0.447·5-s − 2.07·6-s + 0.855·7-s + 0.833·8-s + 0.701·9-s + 0.710·10-s − 0.301·11-s − 1.98·12-s + 0.355·13-s + 1.35·14-s − 0.583·15-s − 0.200·16-s − 0.0338·17-s + 1.11·18-s − 0.229·19-s + 0.681·20-s − 1.11·21-s − 0.479·22-s + 0.689·23-s − 1.08·24-s + 0.200·25-s + 0.564·26-s + 0.389·27-s + 1.30·28-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
| \(L(3)\) |
\(\approx\) |
\(4.374036850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.374036850\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 8.98T + 32T^{2} \) |
| 3 | \( 1 + 20.3T + 243T^{2} \) |
| 7 | \( 1 - 110.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 216.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 40.2T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.20e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−9.317823369554282330480124799620, −8.137968642131671666101164745902, −7.00722384867347766895022362760, −6.20813120183408287702387919649, −5.65255912641788534124807554376, −4.90541694788515460428330409888, −4.38304493685022457117077503540, −3.09164765841954898318286709936, −1.97956679949380965219809146313, −0.76166091751095457880104425851,
0.76166091751095457880104425851, 1.97956679949380965219809146313, 3.09164765841954898318286709936, 4.38304493685022457117077503540, 4.90541694788515460428330409888, 5.65255912641788534124807554376, 6.20813120183408287702387919649, 7.00722384867347766895022362760, 8.137968642131671666101164745902, 9.317823369554282330480124799620
