| L(s) = 1 | + 2-s + 3.24·3-s + 4-s + 1.55·5-s + 3.24·6-s − 3.60·7-s + 8-s + 7.54·9-s + 1.55·10-s − 5.18·11-s + 3.24·12-s − 13-s − 3.60·14-s + 5.04·15-s + 16-s − 5.96·17-s + 7.54·18-s − 19-s + 1.55·20-s − 11.7·21-s − 5.18·22-s + 4.29·23-s + 3.24·24-s − 2.58·25-s − 26-s + 14.7·27-s − 3.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.87·3-s + 0.5·4-s + 0.695·5-s + 1.32·6-s − 1.36·7-s + 0.353·8-s + 2.51·9-s + 0.491·10-s − 1.56·11-s + 0.937·12-s − 0.277·13-s − 0.963·14-s + 1.30·15-s + 0.250·16-s − 1.44·17-s + 1.77·18-s − 0.229·19-s + 0.347·20-s − 2.55·21-s − 1.10·22-s + 0.895·23-s + 0.662·24-s − 0.516·25-s − 0.196·26-s + 2.83·27-s − 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.517772461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.517772461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 9.83T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 0.615T + 61T^{2} \) |
| 67 | \( 1 + 0.987T + 67T^{2} \) |
| 71 | \( 1 + 4.71T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 - 0.789T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−10.53896974605475485788197478097, −10.03542419629717932672372039340, −9.141562304048337911310730887243, −8.349756617551803266920659796152, −7.22843160276578310176244929756, −6.51734864557572031100452977380, −5.09478798309212428707265395498, −3.86969914216570995642031228357, −2.76908488562763858098823260755, −2.32293108381425864428261622317,
2.32293108381425864428261622317, 2.76908488562763858098823260755, 3.86969914216570995642031228357, 5.09478798309212428707265395498, 6.51734864557572031100452977380, 7.22843160276578310176244929756, 8.349756617551803266920659796152, 9.141562304048337911310730887243, 10.03542419629717932672372039340, 10.53896974605475485788197478097
