| L(s) = 1 | + 1.54·2-s − 5.62·4-s + 11.1·5-s + 7·7-s − 21·8-s + 17.2·10-s + 11·11-s − 79.1·13-s + 10.7·14-s + 12.6·16-s − 53.9·17-s + 130.·19-s − 62.7·20-s + 16.9·22-s − 58.5·23-s − 0.331·25-s − 122.·26-s − 39.3·28-s − 277.·29-s + 154.·31-s + 187.·32-s − 83.0·34-s + 78.1·35-s − 297.·37-s + 201.·38-s − 234.·40-s − 304.·41-s + ⋯ |
| L(s) = 1 | + 0.544·2-s − 0.703·4-s + 0.998·5-s + 0.377·7-s − 0.928·8-s + 0.544·10-s + 0.301·11-s − 1.68·13-s + 0.205·14-s + 0.197·16-s − 0.769·17-s + 1.57·19-s − 0.702·20-s + 0.164·22-s − 0.531·23-s − 0.00264·25-s − 0.920·26-s − 0.265·28-s − 1.77·29-s + 0.896·31-s + 1.03·32-s − 0.419·34-s + 0.377·35-s − 1.32·37-s + 0.860·38-s − 0.926·40-s − 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.54T + 8T^{2} \) |
| 5 | \( 1 - 11.1T + 125T^{2} \) |
| 13 | \( 1 + 79.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 304.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 502.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 53.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 263.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 62.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 756.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 135.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 571.T + 9.12e5T^{2} \) |
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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
−9.602036396684893444613730805093, −9.050762787009883833873861241115, −7.86075308491119998922911810810, −6.88670532623123052753168232225, −5.66984460530391837765294128787, −5.16675356874704210567766923999, −4.20706622855885860721122428724, −2.92306714404741700841562893596, −1.72988640033133755240877162168, 0,
1.72988640033133755240877162168, 2.92306714404741700841562893596, 4.20706622855885860721122428724, 5.16675356874704210567766923999, 5.66984460530391837765294128787, 6.88670532623123052753168232225, 7.86075308491119998922911810810, 9.050762787009883833873861241115, 9.602036396684893444613730805093
