| L(s) = 1 | − 1.31·2-s − 0.276·4-s − 3.45·5-s + 7-s + 2.98·8-s + 4.54·10-s − 5.53·13-s − 1.31·14-s − 3.37·16-s + 4.96·17-s + 8.22·19-s + 0.955·20-s − 7.56·23-s + 6.96·25-s + 7.27·26-s − 0.276·28-s − 1.47·29-s + 2.41·31-s − 1.55·32-s − 6.51·34-s − 3.45·35-s − 2.83·37-s − 10.8·38-s − 10.3·40-s + 6.93·41-s + 5.62·43-s + 9.92·46-s + ⋯ |
| L(s) = 1 | − 0.928·2-s − 0.138·4-s − 1.54·5-s + 0.377·7-s + 1.05·8-s + 1.43·10-s − 1.53·13-s − 0.350·14-s − 0.842·16-s + 1.20·17-s + 1.88·19-s + 0.213·20-s − 1.57·23-s + 1.39·25-s + 1.42·26-s − 0.0522·28-s − 0.272·29-s + 0.433·31-s − 0.274·32-s − 1.11·34-s − 0.584·35-s − 0.466·37-s − 1.75·38-s − 1.63·40-s + 1.08·41-s + 0.858·43-s + 1.46·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5273909959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5273909959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 8.22T + 19T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 - 5.62T + 43T^{2} \) |
| 47 | \( 1 + 0.0417T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 - 7.38T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 2.35T + 97T^{2} \) |
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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
−7.77076581929640660299877558786, −7.65789890883389279699581960423, −6.97864948625622996887543188197, −5.63987418043078358375650978233, −4.97651864791401607392111583213, −4.28689474039262266476600228317, −3.61530877827667207897833451144, −2.67498676188235114681332297318, −1.39753097147145910695422218697, −0.45656758419158956590642706413,
0.45656758419158956590642706413, 1.39753097147145910695422218697, 2.67498676188235114681332297318, 3.61530877827667207897833451144, 4.28689474039262266476600228317, 4.97651864791401607392111583213, 5.63987418043078358375650978233, 6.97864948625622996887543188197, 7.65789890883389279699581960423, 7.77076581929640660299877558786
