| L(s) = 1 | − 0.539·2-s − 5.36·3-s − 7.70·4-s + 5·5-s + 2.89·6-s + 3.34·7-s + 8.46·8-s + 1.76·9-s − 2.69·10-s − 11·11-s + 41.3·12-s + 82.0·13-s − 1.80·14-s − 26.8·15-s + 57.1·16-s − 119.·17-s − 0.950·18-s + 19·19-s − 38.5·20-s − 17.9·21-s + 5.92·22-s − 38.3·23-s − 45.4·24-s + 25·25-s − 44.2·26-s + 135.·27-s − 25.7·28-s + ⋯ |
| L(s) = 1 | − 0.190·2-s − 1.03·3-s − 0.963·4-s + 0.447·5-s + 0.196·6-s + 0.180·7-s + 0.374·8-s + 0.0653·9-s − 0.0852·10-s − 0.301·11-s + 0.994·12-s + 1.75·13-s − 0.0344·14-s − 0.461·15-s + 0.892·16-s − 1.70·17-s − 0.0124·18-s + 0.229·19-s − 0.430·20-s − 0.186·21-s + 0.0574·22-s − 0.347·23-s − 0.386·24-s + 0.200·25-s − 0.333·26-s + 0.964·27-s − 0.174·28-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
| \(L(2)\) |
\(\approx\) |
\(0.7675090191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7675090191\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.539T + 8T^{2} \) |
| 3 | \( 1 + 5.36T + 27T^{2} \) |
| 7 | \( 1 - 3.34T + 343T^{2} \) |
| 13 | \( 1 - 82.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 38.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 371.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 258.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 382.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 571.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 717.T + 9.12e5T^{2} \) |
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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
−9.507107851350123295934224453882, −8.683629619365113746228308659297, −8.192712914570056200127613748258, −6.72738894631115466986884142380, −6.08685374381545050878236963916, −5.23026694713789223048778290150, −4.51723955574465809140581992227, −3.39757843055500116991340959113, −1.70906229167744123165543000406, −0.51789676107724966249979817242,
0.51789676107724966249979817242, 1.70906229167744123165543000406, 3.39757843055500116991340959113, 4.51723955574465809140581992227, 5.23026694713789223048778290150, 6.08685374381545050878236963916, 6.72738894631115466986884142380, 8.192712914570056200127613748258, 8.683629619365113746228308659297, 9.507107851350123295934224453882
