| L(s) = 1 | + 2.67·2-s − 0.504·3-s + 5.16·4-s − 5-s − 1.34·6-s + 7-s + 8.47·8-s − 2.74·9-s − 2.67·10-s − 2.60·12-s + 1.47·13-s + 2.67·14-s + 0.504·15-s + 12.3·16-s + 3.78·17-s − 7.35·18-s + 5.96·19-s − 5.16·20-s − 0.504·21-s + 4.26·23-s − 4.27·24-s + 25-s + 3.95·26-s + 2.89·27-s + 5.16·28-s − 8.12·29-s + 1.34·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.291·3-s + 2.58·4-s − 0.447·5-s − 0.550·6-s + 0.377·7-s + 2.99·8-s − 0.915·9-s − 0.846·10-s − 0.751·12-s + 0.409·13-s + 0.715·14-s + 0.130·15-s + 3.09·16-s + 0.917·17-s − 1.73·18-s + 1.36·19-s − 1.15·20-s − 0.110·21-s + 0.890·23-s − 0.872·24-s + 0.200·25-s + 0.775·26-s + 0.557·27-s + 0.976·28-s − 1.50·29-s + 0.246·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.138816075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.138816075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 0.504T + 3T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 6.19T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 2.20T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 - 1.58T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 97 | \( 1 - 7.72T + 97T^{2} \) |
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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
−8.082743489610981615818343979586, −7.35094029249535894629881379340, −6.78477049893854878695490478455, −5.76301398247887843712454350576, −5.39838949115049840875671918994, −4.82923676484284811018262274452, −3.68258061169015492982686650852, −3.35191538817205959920417736605, −2.39034465556301703644960959211, −1.15063305062422092564151444753,
1.15063305062422092564151444753, 2.39034465556301703644960959211, 3.35191538817205959920417736605, 3.68258061169015492982686650852, 4.82923676484284811018262274452, 5.39838949115049840875671918994, 5.76301398247887843712454350576, 6.78477049893854878695490478455, 7.35094029249535894629881379340, 8.082743489610981615818343979586
