| L(s) = 1 | + 5.30·2-s + 7.05·3-s + 20.0·4-s − 5·5-s + 37.4·6-s + 12.7·7-s + 64.0·8-s + 22.8·9-s − 26.5·10-s + 11·11-s + 141.·12-s − 54.1·13-s + 67.3·14-s − 35.2·15-s + 178.·16-s + 67.9·17-s + 120.·18-s + 19·19-s − 100.·20-s + 89.6·21-s + 58.3·22-s + 89.2·23-s + 452.·24-s + 25·25-s − 287.·26-s − 29.5·27-s + 255.·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.35·3-s + 2.51·4-s − 0.447·5-s + 2.54·6-s + 0.685·7-s + 2.83·8-s + 0.844·9-s − 0.838·10-s + 0.301·11-s + 3.41·12-s − 1.15·13-s + 1.28·14-s − 0.607·15-s + 2.79·16-s + 0.969·17-s + 1.58·18-s + 0.229·19-s − 1.12·20-s + 0.931·21-s + 0.564·22-s + 0.808·23-s + 3.84·24-s + 0.200·25-s − 2.16·26-s − 0.210·27-s + 1.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.61571073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.61571073\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 5.30T + 8T^{2} \) |
| 3 | \( 1 - 7.05T + 27T^{2} \) |
| 7 | \( 1 - 12.7T + 343T^{2} \) |
| 13 | \( 1 + 54.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 89.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 69.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 99.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 0.324T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 79.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 355.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 631.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 943.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 276.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 91.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 74.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−9.544449714111642967804619460126, −8.469892421841135461812497511542, −7.45018211335258013936896596945, −7.27182828590645941437305173970, −5.82746285448073091776070985151, −4.95183538962087171195411406357, −4.16468291075641338470067065468, −3.29966453763739036376028816937, −2.62564240744992587220155497102, −1.58645945711525387737873679116,
1.58645945711525387737873679116, 2.62564240744992587220155497102, 3.29966453763739036376028816937, 4.16468291075641338470067065468, 4.95183538962087171195411406357, 5.82746285448073091776070985151, 7.27182828590645941437305173970, 7.45018211335258013936896596945, 8.469892421841135461812497511542, 9.544449714111642967804619460126
