| L(s) = 1 | − 2.37·2-s + 3.63·4-s + 2.08·5-s + 4.17·7-s − 3.87·8-s − 4.95·10-s − 4.20·11-s + 3.62·13-s − 9.90·14-s + 1.92·16-s + 0.150·17-s − 19-s + 7.57·20-s + 9.98·22-s + 3.31·23-s − 0.648·25-s − 8.60·26-s + 15.1·28-s + 0.147·29-s + 5.71·31-s + 3.16·32-s − 0.357·34-s + 8.70·35-s − 1.41·37-s + 2.37·38-s − 8.08·40-s − 4.41·41-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.81·4-s + 0.932·5-s + 1.57·7-s − 1.36·8-s − 1.56·10-s − 1.26·11-s + 1.00·13-s − 2.64·14-s + 0.482·16-s + 0.0365·17-s − 0.229·19-s + 1.69·20-s + 2.12·22-s + 0.690·23-s − 0.129·25-s − 1.68·26-s + 2.86·28-s + 0.0273·29-s + 1.02·31-s + 0.560·32-s − 0.0612·34-s + 1.47·35-s − 0.232·37-s + 0.384·38-s − 1.27·40-s − 0.690·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.341783360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341783360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 4.17T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 0.150T + 17T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 0.147T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 - 0.0827T + 43T^{2} \) |
| 53 | \( 1 + 4.10T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 6.01T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 + 2.81T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 0.569T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−7.941251851454006018961798154460, −7.53330320612815665548223147174, −6.64052177874646788277482585201, −5.90419795434380060414618660463, −5.18055584977251836155721705022, −4.48177569815670261962647614461, −3.06735420809310192836741601404, −2.14381478963293555587456560796, −1.65694036763199667477896827791, −0.77330209971316356461217505268,
0.77330209971316356461217505268, 1.65694036763199667477896827791, 2.14381478963293555587456560796, 3.06735420809310192836741601404, 4.48177569815670261962647614461, 5.18055584977251836155721705022, 5.90419795434380060414618660463, 6.64052177874646788277482585201, 7.53330320612815665548223147174, 7.941251851454006018961798154460
