| L(s) = 1 | + 2.54·2-s − 3-s + 4.46·4-s + 0.329·5-s − 2.54·6-s + 0.595·7-s + 6.27·8-s + 9-s + 0.838·10-s − 11-s − 4.46·12-s + 0.0829·13-s + 1.51·14-s − 0.329·15-s + 7.03·16-s + 6.96·17-s + 2.54·18-s + 1.08·19-s + 1.47·20-s − 0.595·21-s − 2.54·22-s + 3.20·23-s − 6.27·24-s − 4.89·25-s + 0.211·26-s − 27-s + 2.66·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.23·4-s + 0.147·5-s − 1.03·6-s + 0.225·7-s + 2.21·8-s + 0.333·9-s + 0.265·10-s − 0.301·11-s − 1.28·12-s + 0.0230·13-s + 0.404·14-s − 0.0851·15-s + 1.75·16-s + 1.68·17-s + 0.599·18-s + 0.248·19-s + 0.329·20-s − 0.129·21-s − 0.542·22-s + 0.667·23-s − 1.28·24-s − 0.978·25-s + 0.0413·26-s − 0.192·27-s + 0.502·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.928682860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.928682860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 5 | \( 1 - 0.329T + 5T^{2} \) |
| 7 | \( 1 - 0.595T + 7T^{2} \) |
| 13 | \( 1 - 0.0829T + 13T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 - 8.72T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + 5.63T + 37T^{2} \) |
| 41 | \( 1 + 0.0715T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 0.674T + 59T^{2} \) |
| 67 | \( 1 - 0.140T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 + 1.53T + 89T^{2} \) |
| 97 | \( 1 + 8.74T + 97T^{2} \) |
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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
−9.301222561807047332243061006689, −7.945247543384324072987154592194, −7.32233389328453521353776308627, −6.43591453772468082452131288052, −5.67183141143668830859938722649, −5.22547382305447953355841920192, −4.37837106498781826155999563583, −3.48343538284847975671562305552, −2.61418363526040993854397192520, −1.32381880192521783045000275124,
1.32381880192521783045000275124, 2.61418363526040993854397192520, 3.48343538284847975671562305552, 4.37837106498781826155999563583, 5.22547382305447953355841920192, 5.67183141143668830859938722649, 6.43591453772468082452131288052, 7.32233389328453521353776308627, 7.945247543384324072987154592194, 9.301222561807047332243061006689
