| L(s) = 1 | + 5.32·2-s − 3·3-s + 20.3·4-s + 17.7·5-s − 15.9·6-s − 28.1·7-s + 65.7·8-s + 9·9-s + 94.6·10-s − 11·11-s − 61.0·12-s − 35.5·13-s − 149.·14-s − 53.3·15-s + 187.·16-s − 25.4·17-s + 47.9·18-s + 61.7·19-s + 361.·20-s + 84.4·21-s − 58.5·22-s + 135.·23-s − 197.·24-s + 190.·25-s − 189.·26-s − 27·27-s − 572.·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.54·4-s + 1.58·5-s − 1.08·6-s − 1.51·7-s + 2.90·8-s + 0.333·9-s + 2.99·10-s − 0.301·11-s − 1.46·12-s − 0.759·13-s − 2.86·14-s − 0.917·15-s + 2.92·16-s − 0.362·17-s + 0.627·18-s + 0.745·19-s + 4.04·20-s + 0.877·21-s − 0.567·22-s + 1.22·23-s − 1.67·24-s + 1.52·25-s − 1.42·26-s − 0.192·27-s − 3.86·28-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
| \(L(2)\) |
\(\approx\) |
\(8.019062222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.019062222\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.32T + 8T^{2} \) |
| 5 | \( 1 - 17.7T + 125T^{2} \) |
| 7 | \( 1 + 28.1T + 343T^{2} \) |
| 13 | \( 1 + 35.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 271.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 343.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 165.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 519.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 107.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 343.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 47.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 806.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 306.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 303.T + 9.12e5T^{2} \) |
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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
−9.072609096296131106115130247302, −7.33482598961857800006115542448, −6.73649329344759654416537870512, −6.20101348056802676048494931758, −5.52551349567915867309917250887, −5.02210232282631795365541855818, −3.98615064075245017538788948435, −2.73535731733595217831197236144, −2.54415619951176752222363776683, −1.02916953639007589253456945267,
1.02916953639007589253456945267, 2.54415619951176752222363776683, 2.73535731733595217831197236144, 3.98615064075245017538788948435, 5.02210232282631795365541855818, 5.52551349567915867309917250887, 6.20101348056802676048494931758, 6.73649329344759654416537870512, 7.33482598961857800006115542448, 9.072609096296131106115130247302
