| 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
| \(L(1)\) |
\(\approx\) |
\(1.435478045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435478045\) |
| \(L(\frac{3}{2})\) |
|
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
−7.962162657546612034881744597279, −7.32890895064006620692447484270, −6.42741852483826328032053473152, −5.82923072368932658335603478629, −4.96126637986642884003267984026, −4.12571677014584884671948479507, −3.52984701338023764913066230748, −2.85734129872799585831601111258, −1.77959702746595594795625578707, −0.57840194964457342107165103651,
0.57840194964457342107165103651, 1.77959702746595594795625578707, 2.85734129872799585831601111258, 3.52984701338023764913066230748, 4.12571677014584884671948479507, 4.96126637986642884003267984026, 5.82923072368932658335603478629, 6.42741852483826328032053473152, 7.32890895064006620692447484270, 7.962162657546612034881744597279
