| L(s) = 1 | + 2-s − 2.00·3-s + 4-s − 1.79·5-s − 2.00·6-s − 2.43·7-s + 8-s + 1.03·9-s − 1.79·10-s − 1.56·11-s − 2.00·12-s − 4.34·13-s − 2.43·14-s + 3.60·15-s + 16-s − 2.98·17-s + 1.03·18-s − 19-s − 1.79·20-s + 4.89·21-s − 1.56·22-s + 1.92·23-s − 2.00·24-s − 1.78·25-s − 4.34·26-s + 3.95·27-s − 2.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.802·5-s − 0.819·6-s − 0.921·7-s + 0.353·8-s + 0.343·9-s − 0.567·10-s − 0.472·11-s − 0.579·12-s − 1.20·13-s − 0.651·14-s + 0.929·15-s + 0.250·16-s − 0.724·17-s + 0.242·18-s − 0.229·19-s − 0.401·20-s + 1.06·21-s − 0.334·22-s + 0.400·23-s − 0.409·24-s − 0.356·25-s − 0.851·26-s + 0.760·27-s − 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1362080857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1362080857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 23 | \( 1 - 1.92T + 23T^{2} \) |
| 29 | \( 1 - 0.562T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 8.05T + 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 + 2.99T + 79T^{2} \) |
| 83 | \( 1 - 9.69T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 3.44T + 97T^{2} \) |
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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
−7.58165345197582289914601848416, −6.77847613740987211125871291954, −6.58326284974081863155553367799, −5.61132650021232063664217848759, −5.03489600002570827468250383593, −4.52230491609037296156401715074, −3.53625979850250200573127002415, −2.93885785142970422991897240146, −1.84793415044383026942513430377, −0.16246472425579553278270564490,
0.16246472425579553278270564490, 1.84793415044383026942513430377, 2.93885785142970422991897240146, 3.53625979850250200573127002415, 4.52230491609037296156401715074, 5.03489600002570827468250383593, 5.61132650021232063664217848759, 6.58326284974081863155553367799, 6.77847613740987211125871291954, 7.58165345197582289914601848416
