| L(s) = 1 | − 0.275·2-s − 3·3-s − 7.92·4-s + 15.4·5-s + 0.825·6-s − 21.6·7-s + 4.38·8-s + 9·9-s − 4.25·10-s + 11·11-s + 23.7·12-s + 80.4·13-s + 5.95·14-s − 46.3·15-s + 62.1·16-s − 115.·17-s − 2.47·18-s − 0.685·19-s − 122.·20-s + 64.9·21-s − 3.02·22-s + 74.6·23-s − 13.1·24-s + 114.·25-s − 22.1·26-s − 27·27-s + 171.·28-s + ⋯ |
| L(s) = 1 | − 0.0972·2-s − 0.577·3-s − 0.990·4-s + 1.38·5-s + 0.0561·6-s − 1.16·7-s + 0.193·8-s + 0.333·9-s − 0.134·10-s + 0.301·11-s + 0.571·12-s + 1.71·13-s + 0.113·14-s − 0.798·15-s + 0.971·16-s − 1.64·17-s − 0.0324·18-s − 0.00827·19-s − 1.37·20-s + 0.674·21-s − 0.0293·22-s + 0.677·23-s − 0.111·24-s + 0.913·25-s − 0.166·26-s − 0.192·27-s + 1.15·28-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
| \(L(2)\) |
\(\approx\) |
\(1.461512684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461512684\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.275T + 8T^{2} \) |
| 5 | \( 1 - 15.4T + 125T^{2} \) |
| 7 | \( 1 + 21.6T + 343T^{2} \) |
| 13 | \( 1 - 80.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.685T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.01T + 2.43e4T^{2} \) |
| 31 | \( 1 - 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 181.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 329.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 580.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 370.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 692.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 691.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 442.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 393.T + 9.12e5T^{2} \) |
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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
−9.029892768931392065126151530096, −8.331013728759751036239925157263, −6.86522867231544972612048473107, −6.23139300632943258228290030495, −5.86098503755811667039421217850, −4.79433676334205713517868410955, −3.97574152948015881956078786921, −2.92702660997788367049358218453, −1.58609027094524430790541258850, −0.61549747153666215951501678248,
0.61549747153666215951501678248, 1.58609027094524430790541258850, 2.92702660997788367049358218453, 3.97574152948015881956078786921, 4.79433676334205713517868410955, 5.86098503755811667039421217850, 6.23139300632943258228290030495, 6.86522867231544972612048473107, 8.331013728759751036239925157263, 9.029892768931392065126151530096
