| L(s) = 1 | + 3.32·5-s + 2.77·7-s + 2.53·11-s + 0.745·13-s − 4.83·17-s − 19-s − 1.29·23-s + 6.07·25-s + 2.01·29-s + 1.93·31-s + 9.23·35-s − 1.83·37-s + 11.7·41-s + 5.64·43-s + 4.39·47-s + 0.707·49-s + 1.00·53-s + 8.42·55-s − 0.602·59-s + 4.18·61-s + 2.48·65-s + 1.92·67-s − 0.548·71-s − 5.91·73-s + 7.03·77-s + 8.96·79-s − 11.0·83-s + ⋯ |
| L(s) = 1 | + 1.48·5-s + 1.04·7-s + 0.763·11-s + 0.206·13-s − 1.17·17-s − 0.229·19-s − 0.270·23-s + 1.21·25-s + 0.373·29-s + 0.348·31-s + 1.56·35-s − 0.300·37-s + 1.84·41-s + 0.860·43-s + 0.641·47-s + 0.101·49-s + 0.138·53-s + 1.13·55-s − 0.0784·59-s + 0.535·61-s + 0.307·65-s + 0.235·67-s − 0.0651·71-s − 0.692·73-s + 0.801·77-s + 1.00·79-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.169106530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.169106530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 0.745T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 - 1.93T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 0.602T + 59T^{2} \) |
| 61 | \( 1 - 4.18T + 61T^{2} \) |
| 67 | \( 1 - 1.92T + 67T^{2} \) |
| 71 | \( 1 + 0.548T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 8.96T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.631566671273619243176850092116, −7.71943837068733766637805166813, −6.80738783696948696485592962060, −6.16915916182906053376365435420, −5.56541430144458939066321437127, −4.67120044302587926536068850220, −4.04429321560936542215025122956, −2.61506083132148675794828405034, −1.96825789614456359118567323803, −1.10775981562934953987320481612,
1.10775981562934953987320481612, 1.96825789614456359118567323803, 2.61506083132148675794828405034, 4.04429321560936542215025122956, 4.67120044302587926536068850220, 5.56541430144458939066321437127, 6.16915916182906053376365435420, 6.80738783696948696485592962060, 7.71943837068733766637805166813, 8.631566671273619243176850092116
