| L(s) = 1 | − 0.158·2-s − 3·3-s − 7.97·4-s + 7.59·5-s + 0.476·6-s − 16.2·7-s + 2.53·8-s + 9·9-s − 1.20·10-s − 11·11-s + 23.9·12-s + 56.3·13-s + 2.57·14-s − 22.7·15-s + 63.3·16-s + 54.6·17-s − 1.43·18-s − 148.·19-s − 60.5·20-s + 48.6·21-s + 1.74·22-s − 148.·23-s − 7.61·24-s − 67.3·25-s − 8.95·26-s − 27·27-s + 129.·28-s + ⋯ |
| L(s) = 1 | − 0.0561·2-s − 0.577·3-s − 0.996·4-s + 0.679·5-s + 0.0324·6-s − 0.875·7-s + 0.112·8-s + 0.333·9-s − 0.0381·10-s − 0.301·11-s + 0.575·12-s + 1.20·13-s + 0.0491·14-s − 0.392·15-s + 0.990·16-s + 0.779·17-s − 0.0187·18-s − 1.79·19-s − 0.677·20-s + 0.505·21-s + 0.0169·22-s − 1.34·23-s − 0.0647·24-s − 0.538·25-s − 0.0675·26-s − 0.192·27-s + 0.872·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 0.158T + 8T^{2} \) |
| 5 | \( 1 - 7.59T + 125T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 13 | \( 1 - 56.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 7.15T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 247.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 16.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 371.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 68.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 788.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 903.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 630.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 611.T + 9.12e5T^{2} \) |
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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
−8.444624696042365225399214243622, −7.82219281517849307537070215521, −6.37372640876483924891703435455, −6.13771715841699807688647295225, −5.33016491864093104873557257404, −4.23617656303875998416144317579, −3.64869459788546931503962303248, −2.28285064547814847405803485854, −1.00698842434473976955352137740, 0,
1.00698842434473976955352137740, 2.28285064547814847405803485854, 3.64869459788546931503962303248, 4.23617656303875998416144317579, 5.33016491864093104873557257404, 6.13771715841699807688647295225, 6.37372640876483924891703435455, 7.82219281517849307537070215521, 8.444624696042365225399214243622
