| L(s) = 1 | − 2·5-s − 7-s + 2·11-s + 5.12·13-s + 5.12·17-s − 19-s + 1.12·23-s − 25-s + 0.876·29-s + 7.12·31-s + 2·35-s − 9.12·37-s + 8.24·41-s + 4·43-s + 49-s − 3.12·53-s − 4·55-s − 10.2·59-s − 2·61-s − 10.2·65-s + 2.24·67-s + 15.3·71-s − 10·73-s − 2·77-s + 4.87·79-s − 0.876·83-s − 10.2·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s + 1.42·13-s + 1.24·17-s − 0.229·19-s + 0.234·23-s − 0.200·25-s + 0.162·29-s + 1.27·31-s + 0.338·35-s − 1.49·37-s + 1.28·41-s + 0.609·43-s + 0.142·49-s − 0.428·53-s − 0.539·55-s − 1.33·59-s − 0.256·61-s − 1.27·65-s + 0.274·67-s + 1.82·71-s − 1.17·73-s − 0.227·77-s + 0.548·79-s − 0.0962·83-s − 1.11·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.873305258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.873305258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 0.876T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 9.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−7.84951657561159274106672978736, −6.99337560207161330067812098486, −6.31480450206719184619501330049, −5.79645148908963802216614021230, −4.84588209817386277044704354395, −3.97319549997191632117540244165, −3.58789587897799903431276403506, −2.82970041661384375884560329137, −1.52196817292305346432031620228, −0.70153349473876257868841530637,
0.70153349473876257868841530637, 1.52196817292305346432031620228, 2.82970041661384375884560329137, 3.58789587897799903431276403506, 3.97319549997191632117540244165, 4.84588209817386277044704354395, 5.79645148908963802216614021230, 6.31480450206719184619501330049, 6.99337560207161330067812098486, 7.84951657561159274106672978736
