| L(s) = 1 | − 0.515·3-s + 2.19·5-s − 2.73·9-s + 4.22·11-s + 3.63·13-s − 1.13·15-s − 7.98·17-s + 1.14·19-s − 7.43·23-s − 0.161·25-s + 2.95·27-s + 6.08·29-s − 1.33·31-s − 2.17·33-s − 2.87·37-s − 1.87·39-s − 41-s + 5.14·43-s − 6.01·45-s + 8.12·47-s + 4.11·51-s − 1.63·53-s + 9.28·55-s − 0.587·57-s − 2.97·59-s + 10.2·61-s + 8.00·65-s + ⋯ |
| L(s) = 1 | − 0.297·3-s + 0.983·5-s − 0.911·9-s + 1.27·11-s + 1.00·13-s − 0.292·15-s − 1.93·17-s + 0.261·19-s − 1.55·23-s − 0.0323·25-s + 0.568·27-s + 1.12·29-s − 0.239·31-s − 0.378·33-s − 0.473·37-s − 0.300·39-s − 0.156·41-s + 0.784·43-s − 0.896·45-s + 1.18·47-s + 0.576·51-s − 0.224·53-s + 1.25·55-s − 0.0777·57-s − 0.387·59-s + 1.31·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.104337354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104337354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.515T + 3T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 + 7.98T + 17T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 + 1.33T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 2.46T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 - 7.51T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.975122720028078222677779118023, −6.74442308728575694725104691013, −6.40718617017307275817702938344, −5.92913866881740980560924495202, −5.18290986678522187825240996845, −4.21900203232766503407233534210, −3.64246626454295351779814827612, −2.44399044242890942171285233338, −1.87180961663245823767535219679, −0.72695519637306173515390691407,
0.72695519637306173515390691407, 1.87180961663245823767535219679, 2.44399044242890942171285233338, 3.64246626454295351779814827612, 4.21900203232766503407233534210, 5.18290986678522187825240996845, 5.92913866881740980560924495202, 6.40718617017307275817702938344, 6.74442308728575694725104691013, 7.975122720028078222677779118023
