| L(s) = 1 | − 5.51·2-s + 3·3-s + 22.3·4-s − 14.2·5-s − 16.5·6-s + 3.43·7-s − 79.1·8-s + 9·9-s + 78.4·10-s + 11·11-s + 67.0·12-s − 2.00·13-s − 18.9·14-s − 42.7·15-s + 257.·16-s − 9.45·17-s − 49.5·18-s + 151.·19-s − 318.·20-s + 10.3·21-s − 60.6·22-s − 109.·23-s − 237.·24-s + 77.7·25-s + 11.0·26-s + 27·27-s + 76.9·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.79·4-s − 1.27·5-s − 1.12·6-s + 0.185·7-s − 3.49·8-s + 0.333·9-s + 2.48·10-s + 0.301·11-s + 1.61·12-s − 0.0426·13-s − 0.361·14-s − 0.735·15-s + 4.02·16-s − 0.134·17-s − 0.649·18-s + 1.82·19-s − 3.56·20-s + 0.107·21-s − 0.587·22-s − 0.996·23-s − 2.01·24-s + 0.621·25-s + 0.0831·26-s + 0.192·27-s + 0.519·28-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 | \( 1 + 5.51T + 8T^{2} \) |
| 5 | \( 1 + 14.2T + 125T^{2} \) |
| 7 | \( 1 - 3.43T + 343T^{2} \) |
| 13 | \( 1 + 2.00T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.45T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 75.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 562.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 142.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 220.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 84.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−8.363850031546220674722195722029, −7.80199488306511633076162787991, −7.34364434361250691075502037357, −6.60982308758460280721997598592, −5.39723078379573740767680068920, −3.84955820229746410396464182078, −3.14938708006563969994882659249, −2.02052045007240309065834296705, −1.02151376612087870116059231705, 0,
1.02151376612087870116059231705, 2.02052045007240309065834296705, 3.14938708006563969994882659249, 3.84955820229746410396464182078, 5.39723078379573740767680068920, 6.60982308758460280721997598592, 7.34364434361250691075502037357, 7.80199488306511633076162787991, 8.363850031546220674722195722029
