L(s) = 1 | + 3-s + 0.624·5-s + 0.240·7-s + 9-s − 5.05·11-s + 4.23·13-s + 0.624·15-s + 1.70·17-s − 5.35·19-s + 0.240·21-s + 3.26·23-s − 4.60·25-s + 27-s − 8.98·29-s + 2.00·31-s − 5.05·33-s + 0.150·35-s − 5.86·37-s + 4.23·39-s − 10.7·41-s − 2.48·43-s + 0.624·45-s + 2.29·47-s − 6.94·49-s + 1.70·51-s + 10.5·53-s − 3.15·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.279·5-s + 0.0908·7-s + 0.333·9-s − 1.52·11-s + 1.17·13-s + 0.161·15-s + 0.413·17-s − 1.22·19-s + 0.0524·21-s + 0.680·23-s − 0.921·25-s + 0.192·27-s − 1.66·29-s + 0.360·31-s − 0.879·33-s + 0.0253·35-s − 0.964·37-s + 0.677·39-s − 1.68·41-s − 0.378·43-s + 0.0931·45-s + 0.334·47-s − 0.991·49-s + 0.238·51-s + 1.45·53-s − 0.425·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
good | 5 | \( 1 - 0.624T + 5T^{2} \) |
| 7 | \( 1 - 0.240T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 + 8.98T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.19T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7.05T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.86567997862821912942371879450, −7.12595225876339536339161782917, −6.28206029051796513541740594581, −5.54810867194274606566079507617, −4.90760493692018921773293001210, −3.86718238913730281453052872063, −3.25609829871814992049119108372, −2.28029655521994359855697968402, −1.56499410562770744351001599484, 0,
1.56499410562770744351001599484, 2.28029655521994359855697968402, 3.25609829871814992049119108372, 3.86718238913730281453052872063, 4.90760493692018921773293001210, 5.54810867194274606566079507617, 6.28206029051796513541740594581, 7.12595225876339536339161782917, 7.86567997862821912942371879450