| L(s) = 1 | − 0.389·2-s − 3-s − 1.84·4-s + 0.642·5-s + 0.389·6-s − 1.35·7-s + 1.49·8-s + 9-s − 0.250·10-s − 3.00·11-s + 1.84·12-s − 13-s + 0.527·14-s − 0.642·15-s + 3.11·16-s + 3.71·17-s − 0.389·18-s − 2.28·19-s − 1.18·20-s + 1.35·21-s + 1.16·22-s + 1.06·23-s − 1.49·24-s − 4.58·25-s + 0.389·26-s − 27-s + 2.50·28-s + ⋯ |
| L(s) = 1 | − 0.275·2-s − 0.577·3-s − 0.924·4-s + 0.287·5-s + 0.158·6-s − 0.511·7-s + 0.529·8-s + 0.333·9-s − 0.0791·10-s − 0.905·11-s + 0.533·12-s − 0.277·13-s + 0.140·14-s − 0.165·15-s + 0.778·16-s + 0.901·17-s − 0.0917·18-s − 0.523·19-s − 0.265·20-s + 0.295·21-s + 0.249·22-s + 0.222·23-s − 0.305·24-s − 0.917·25-s + 0.0763·26-s − 0.192·27-s + 0.473·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6100395914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6100395914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.389T + 2T^{2} \) |
| 5 | \( 1 - 0.642T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 + 0.297T + 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 9.99T + 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 - 1.23T + 67T^{2} \) |
| 71 | \( 1 + 1.30T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.32T + 97T^{2} \) |
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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
−8.233058865545657636856888005003, −7.972870319964275815416626723055, −6.98060848002871931729947720410, −6.12461250992668290230478109791, −5.40732773535908164079851783260, −4.85145652652512725305419697434, −3.92688881640637528757653699669, −3.02955898114280132760674227989, −1.76787969253429963014561535713, −0.48062017107759650270416384461,
0.48062017107759650270416384461, 1.76787969253429963014561535713, 3.02955898114280132760674227989, 3.92688881640637528757653699669, 4.85145652652512725305419697434, 5.40732773535908164079851783260, 6.12461250992668290230478109791, 6.98060848002871931729947720410, 7.972870319964275815416626723055, 8.233058865545657636856888005003
