| L(s) = 1 | − 1.36·2-s − 0.139·4-s + 2.07·5-s − 0.103·7-s + 2.91·8-s − 2.83·10-s − 6.62·13-s + 0.141·14-s − 3.70·16-s + 3.79·17-s + 6.60·19-s − 0.289·20-s + 2.58·23-s − 0.684·25-s + 9.03·26-s + 0.0144·28-s + 9.95·29-s − 2.20·31-s − 0.786·32-s − 5.17·34-s − 0.215·35-s + 3.26·37-s − 9.00·38-s + 6.06·40-s − 1.43·41-s + 3.82·43-s − 3.52·46-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.0696·4-s + 0.929·5-s − 0.0392·7-s + 1.03·8-s − 0.896·10-s − 1.83·13-s + 0.0378·14-s − 0.925·16-s + 0.920·17-s + 1.51·19-s − 0.0647·20-s + 0.538·23-s − 0.136·25-s + 1.77·26-s + 0.00273·28-s + 1.84·29-s − 0.395·31-s − 0.139·32-s − 0.887·34-s − 0.0364·35-s + 0.535·37-s − 1.46·38-s + 0.958·40-s − 0.223·41-s + 0.582·43-s − 0.519·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294297112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294297112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 0.103T + 7T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 9.95T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 - 3.43T + 47T^{2} \) |
| 53 | \( 1 + 0.0632T + 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 + 7.99T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−7.54061538768663009650892756655, −7.37446668011529425669244152210, −6.45058047599106051082885272438, −5.49162422655860744798819415667, −5.06807197466911154563182679853, −4.36731414699319706026892567233, −3.14589816409612208014592806675, −2.45564816636967372510970754265, −1.48843872200683593886139480997, −0.67154693567210396758398779667,
0.67154693567210396758398779667, 1.48843872200683593886139480997, 2.45564816636967372510970754265, 3.14589816409612208014592806675, 4.36731414699319706026892567233, 5.06807197466911154563182679853, 5.49162422655860744798819415667, 6.45058047599106051082885272438, 7.37446668011529425669244152210, 7.54061538768663009650892756655
