| L(s) = 1 | + 2-s + 2.05·3-s + 4-s − 3.09·5-s + 2.05·6-s + 3.17·7-s + 8-s + 1.22·9-s − 3.09·10-s + 1.25·11-s + 2.05·12-s + 1.46·13-s + 3.17·14-s − 6.35·15-s + 16-s + 6.42·17-s + 1.22·18-s + 2.45·19-s − 3.09·20-s + 6.52·21-s + 1.25·22-s + 3.00·23-s + 2.05·24-s + 4.56·25-s + 1.46·26-s − 3.65·27-s + 3.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s − 1.38·5-s + 0.838·6-s + 1.19·7-s + 0.353·8-s + 0.407·9-s − 0.977·10-s + 0.378·11-s + 0.593·12-s + 0.407·13-s + 0.847·14-s − 1.64·15-s + 0.250·16-s + 1.55·17-s + 0.288·18-s + 0.563·19-s − 0.691·20-s + 1.42·21-s + 0.267·22-s + 0.626·23-s + 0.419·24-s + 0.912·25-s + 0.288·26-s − 0.702·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.878418974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.878418974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 6.42T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 + 7.75T + 29T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 + 0.502T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 0.170T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
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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
−8.066329320553783621222383670098, −7.48763724843903897745149935886, −7.01673303525743790419691177896, −5.65430583579075409113684236314, −5.14649940034836248375250492423, −4.10689890506938916589031388734, −3.68001022677000646770247812495, −3.11351708220748392466990267388, −2.03764845681864355100927569534, −1.05580158838053026801291676741,
1.05580158838053026801291676741, 2.03764845681864355100927569534, 3.11351708220748392466990267388, 3.68001022677000646770247812495, 4.10689890506938916589031388734, 5.14649940034836248375250492423, 5.65430583579075409113684236314, 7.01673303525743790419691177896, 7.48763724843903897745149935886, 8.066329320553783621222383670098
