| L(s) = 1 | − 2-s + 2.76·3-s + 4-s − 2.60·5-s − 2.76·6-s − 3.00·7-s − 8-s + 4.64·9-s + 2.60·10-s + 4.33·11-s + 2.76·12-s − 0.432·13-s + 3.00·14-s − 7.20·15-s + 16-s + 1.52·17-s − 4.64·18-s + 5.33·19-s − 2.60·20-s − 8.31·21-s − 4.33·22-s − 23-s − 2.76·24-s + 1.79·25-s + 0.432·26-s + 4.54·27-s − 3.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.59·3-s + 0.5·4-s − 1.16·5-s − 1.12·6-s − 1.13·7-s − 0.353·8-s + 1.54·9-s + 0.824·10-s + 1.30·11-s + 0.798·12-s − 0.120·13-s + 0.803·14-s − 1.86·15-s + 0.250·16-s + 0.369·17-s − 1.09·18-s + 1.22·19-s − 0.582·20-s − 1.81·21-s − 0.925·22-s − 0.208·23-s − 0.564·24-s + 0.358·25-s + 0.0848·26-s + 0.874·27-s − 0.568·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923476511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923476511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 0.432T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 + 0.502T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 - 0.273T + 53T^{2} \) |
| 59 | \( 1 + 2.64T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 - 8.70T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 6.18T + 97T^{2} \) |
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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
−8.038982565135251009960561133340, −7.73240574811784951648640386904, −6.72883903634291393272280314208, −6.56940192578402130726095182873, −5.06218776336867955371066886580, −3.92018519737605064384331903847, −3.43704600029987613630587658350, −3.03111762918474803953371549659, −1.86518181515476755574815958952, −0.75425457321401128738739545253,
0.75425457321401128738739545253, 1.86518181515476755574815958952, 3.03111762918474803953371549659, 3.43704600029987613630587658350, 3.92018519737605064384331903847, 5.06218776336867955371066886580, 6.56940192578402130726095182873, 6.72883903634291393272280314208, 7.73240574811784951648640386904, 8.038982565135251009960561133340
