| L(s) = 1 | + 2.56·2-s − 6.68·3-s − 1.43·4-s − 9.31·5-s − 17.1·6-s + 0.123·7-s − 24.1·8-s + 17.6·9-s − 23.8·10-s + 8.56·11-s + 9.61·12-s − 28.9·13-s + 0.315·14-s + 62.2·15-s − 50.4·16-s + 78.3·17-s + 45.3·18-s + 2.16·19-s + 13.3·20-s − 0.822·21-s + 21.9·22-s − 97.4·23-s + 161.·24-s − 38.2·25-s − 74.1·26-s + 62.2·27-s − 0.177·28-s + ⋯ |
| L(s) = 1 | + 0.905·2-s − 1.28·3-s − 0.179·4-s − 0.833·5-s − 1.16·6-s + 0.00664·7-s − 1.06·8-s + 0.654·9-s − 0.754·10-s + 0.234·11-s + 0.231·12-s − 0.617·13-s + 0.00601·14-s + 1.07·15-s − 0.787·16-s + 1.11·17-s + 0.593·18-s + 0.0261·19-s + 0.149·20-s − 0.00855·21-s + 0.212·22-s − 0.883·23-s + 1.37·24-s − 0.305·25-s − 0.559·26-s + 0.443·27-s − 0.00119·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 3 | \( 1 + 6.68T + 27T^{2} \) |
| 5 | \( 1 + 9.31T + 125T^{2} \) |
| 7 | \( 1 - 0.123T + 343T^{2} \) |
| 11 | \( 1 - 8.56T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.16T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 279.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 70.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 106.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 91.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 631.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 527.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 165.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−14.09009569684376407952327667369, −12.59323266426905122381893413816, −12.03463161816159417681930726310, −11.10337459594264478293291732953, −9.547059276633570034796313037630, −7.72516514523608042367792184189, −6.09408988766832057563390579686, −5.07285028894966301457082191587, −3.75340164446493022369092648062, 0,
3.75340164446493022369092648062, 5.07285028894966301457082191587, 6.09408988766832057563390579686, 7.72516514523608042367792184189, 9.547059276633570034796313037630, 11.10337459594264478293291732953, 12.03463161816159417681930726310, 12.59323266426905122381893413816, 14.09009569684376407952327667369
