| L(s) = 1 | − 3-s − 0.244·5-s + 2.69·7-s + 9-s + 4.17·11-s − 4.50·13-s + 0.244·15-s + 1.19·17-s + 2.55·19-s − 2.69·21-s + 4.82·23-s − 4.94·25-s − 27-s − 2.33·29-s + 8.83·31-s − 4.17·33-s − 0.659·35-s + 6.02·37-s + 4.50·39-s − 9.45·41-s + 7.71·43-s − 0.244·45-s + 2.80·47-s + 0.264·49-s − 1.19·51-s + 3.72·53-s − 1.02·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.109·5-s + 1.01·7-s + 0.333·9-s + 1.25·11-s − 1.24·13-s + 0.0632·15-s + 0.290·17-s + 0.586·19-s − 0.588·21-s + 1.00·23-s − 0.988·25-s − 0.192·27-s − 0.433·29-s + 1.58·31-s − 0.726·33-s − 0.111·35-s + 0.989·37-s + 0.721·39-s − 1.47·41-s + 1.17·43-s − 0.0364·45-s + 0.408·47-s + 0.0378·49-s − 0.167·51-s + 0.511·53-s − 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966071425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966071425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 0.244T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 - 8.83T + 31T^{2} \) |
| 37 | \( 1 - 6.02T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 2.80T + 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 1.84T + 79T^{2} \) |
| 83 | \( 1 + 6.03T + 83T^{2} \) |
| 89 | \( 1 + 0.866T + 89T^{2} \) |
| 97 | \( 1 - 7.72T + 97T^{2} \) |
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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
−7.906842607619735278195857654340, −7.39475798789342869257365699279, −6.69444079354327610490915259736, −5.90994096966200811241316137058, −5.10385982472117631644815726315, −4.58511115815435098012842036657, −3.83064463014156521437524700271, −2.71674446453759314120083286921, −1.67357902885938297346389867964, −0.806258213282553919356179206094,
0.806258213282553919356179206094, 1.67357902885938297346389867964, 2.71674446453759314120083286921, 3.83064463014156521437524700271, 4.58511115815435098012842036657, 5.10385982472117631644815726315, 5.90994096966200811241316137058, 6.69444079354327610490915259736, 7.39475798789342869257365699279, 7.906842607619735278195857654340
