| L(s) = 1 | + 1.47·3-s + 2.24·5-s + 2.24·7-s − 0.837·9-s − 0.249·11-s − 2.77·13-s + 3.30·15-s + 4.24·17-s + 8.61·19-s + 3.30·21-s + 23-s + 0.0586·25-s − 5.64·27-s − 4.66·29-s − 4.08·31-s − 0.366·33-s + 5.05·35-s − 8.86·37-s − 4.08·39-s − 1.22·41-s + 7.55·43-s − 1.88·45-s − 1.96·47-s − 1.94·49-s + 6.24·51-s + 3.43·53-s − 0.560·55-s + ⋯ |
| L(s) = 1 | + 0.849·3-s + 1.00·5-s + 0.850·7-s − 0.279·9-s − 0.0751·11-s − 0.770·13-s + 0.854·15-s + 1.03·17-s + 1.97·19-s + 0.721·21-s + 0.208·23-s + 0.0117·25-s − 1.08·27-s − 0.865·29-s − 0.733·31-s − 0.0637·33-s + 0.855·35-s − 1.45·37-s − 0.654·39-s − 0.190·41-s + 1.15·43-s − 0.280·45-s − 0.287·47-s − 0.277·49-s + 0.875·51-s + 0.472·53-s − 0.0755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.473810686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.473810686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 0.249T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 - 8.61T + 19T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 3.68T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 2.86T + 97T^{2} \) |
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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
−10.05249859072722905440545813040, −9.562554182338264452260867074999, −8.741264010013746172728157919380, −7.78694597941709057813828777675, −7.18424038619421924713035633100, −5.52950758382450901817571538457, −5.31913087447083659485689200554, −3.65270918327833642464046898211, −2.61542815313857160576671531883, −1.55955941695692639218879590501,
1.55955941695692639218879590501, 2.61542815313857160576671531883, 3.65270918327833642464046898211, 5.31913087447083659485689200554, 5.52950758382450901817571538457, 7.18424038619421924713035633100, 7.78694597941709057813828777675, 8.741264010013746172728157919380, 9.562554182338264452260867074999, 10.05249859072722905440545813040
