| L(s) = 1 | − 2-s + 0.549·3-s + 4-s − 0.696·5-s − 0.549·6-s − 3.39·7-s − 8-s − 2.69·9-s + 0.696·10-s + 2.65·11-s + 0.549·12-s − 4.55·13-s + 3.39·14-s − 0.382·15-s + 16-s − 7.01·17-s + 2.69·18-s − 19-s − 0.696·20-s − 1.86·21-s − 2.65·22-s + 0.400·23-s − 0.549·24-s − 4.51·25-s + 4.55·26-s − 3.12·27-s − 3.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.317·3-s + 0.5·4-s − 0.311·5-s − 0.224·6-s − 1.28·7-s − 0.353·8-s − 0.899·9-s + 0.220·10-s + 0.800·11-s + 0.158·12-s − 1.26·13-s + 0.908·14-s − 0.0987·15-s + 0.250·16-s − 1.70·17-s + 0.635·18-s − 0.229·19-s − 0.155·20-s − 0.407·21-s − 0.566·22-s + 0.0834·23-s − 0.112·24-s − 0.903·25-s + 0.893·26-s − 0.602·27-s − 0.642·28-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
| \(L(1)\) |
\(\approx\) |
\(0.09079631131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09079631131\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.549T + 3T^{2} \) |
| 5 | \( 1 + 0.696T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 7.01T + 17T^{2} \) |
| 23 | \( 1 - 0.400T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 - 7.44T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 - 0.980T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.82754245922475096858404402356, −7.11712816467343340379265194854, −6.69903665300945710664792850725, −5.93173103690196980810184744341, −5.17107423839064908473436008372, −3.95545492068017179893633630762, −3.50907504940511354419190740881, −2.49021057913721279641392094251, −1.94909188716858172481938475096, −0.14908248489389702812136090878,
0.14908248489389702812136090878, 1.94909188716858172481938475096, 2.49021057913721279641392094251, 3.50907504940511354419190740881, 3.95545492068017179893633630762, 5.17107423839064908473436008372, 5.93173103690196980810184744341, 6.69903665300945710664792850725, 7.11712816467343340379265194854, 7.82754245922475096858404402356
