| L(s) = 1 | + 4.77·2-s + 4.58·3-s + 14.7·4-s − 9.06·5-s + 21.8·6-s − 11.8·7-s + 32.3·8-s − 5.97·9-s − 43.2·10-s + 67.7·12-s − 13·13-s − 56.7·14-s − 41.5·15-s + 36.2·16-s + 15.1·17-s − 28.5·18-s − 32.9·19-s − 134.·20-s − 54.5·21-s + 77.8·23-s + 148.·24-s − 42.7·25-s − 62.0·26-s − 151.·27-s − 175.·28-s − 91.3·29-s − 198.·30-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.882·3-s + 1.84·4-s − 0.810·5-s + 1.48·6-s − 0.642·7-s + 1.43·8-s − 0.221·9-s − 1.36·10-s + 1.63·12-s − 0.277·13-s − 1.08·14-s − 0.715·15-s + 0.567·16-s + 0.216·17-s − 0.373·18-s − 0.397·19-s − 1.49·20-s − 0.566·21-s + 0.705·23-s + 1.26·24-s − 0.342·25-s − 0.468·26-s − 1.07·27-s − 1.18·28-s − 0.584·29-s − 1.20·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 4.77T + 8T^{2} \) |
| 3 | \( 1 - 4.58T + 27T^{2} \) |
| 5 | \( 1 + 9.06T + 125T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 17 | \( 1 - 15.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 50.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 72.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 324.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 333.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 881.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 132.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 448.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 970.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 632.T + 9.12e5T^{2} \) |
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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
−8.545963817227158257525598497371, −7.65077029882895275426051315978, −6.95336782409179947893094058295, −6.06000140331192785049886508687, −5.20674736754156148449383838334, −4.24636631934555664874985829644, −3.48370388675584054212807208382, −3.00746971872333484207209006877, −1.98646351506393672725992097518, 0,
1.98646351506393672725992097518, 3.00746971872333484207209006877, 3.48370388675584054212807208382, 4.24636631934555664874985829644, 5.20674736754156148449383838334, 6.06000140331192785049886508687, 6.95336782409179947893094058295, 7.65077029882895275426051315978, 8.545963817227158257525598497371
