| L(s) = 1 | + 2.30·3-s + 0.631·5-s − 0.267·7-s + 2.30·9-s + 3.34·11-s + 5.71·13-s + 1.45·15-s + 3.40·17-s − 19-s − 0.615·21-s + 3.34·23-s − 4.60·25-s − 1.59·27-s + 0.469·29-s − 3.82·31-s + 7.70·33-s − 0.168·35-s − 6.35·37-s + 13.1·39-s + 11.1·41-s + 6.63·43-s + 1.45·45-s + 3.70·47-s − 6.92·49-s + 7.83·51-s − 53-s + 2.11·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.282·5-s − 0.101·7-s + 0.768·9-s + 1.00·11-s + 1.58·13-s + 0.375·15-s + 0.824·17-s − 0.229·19-s − 0.134·21-s + 0.696·23-s − 0.920·25-s − 0.307·27-s + 0.0872·29-s − 0.687·31-s + 1.34·33-s − 0.0285·35-s − 1.04·37-s + 2.10·39-s + 1.73·41-s + 1.01·43-s + 0.217·45-s + 0.539·47-s − 0.989·49-s + 1.09·51-s − 0.137·53-s + 0.284·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.839965024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839965024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 0.631T + 5T^{2} \) |
| 7 | \( 1 + 0.267T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 0.469T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 6.35T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 59 | \( 1 - 0.818T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 + 9.25T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 - 7.98T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.602238465302687272981246570171, −7.81206313032829605943361950093, −7.17307885297614968893280201954, −6.16289593776083308319222379998, −5.69544589779322181708120639719, −4.33329976538363918399281662247, −3.66010067652925407878942687516, −3.10307466441664728067552194610, −1.99072426745787200166802887959, −1.16828550475223654843246664227,
1.16828550475223654843246664227, 1.99072426745787200166802887959, 3.10307466441664728067552194610, 3.66010067652925407878942687516, 4.33329976538363918399281662247, 5.69544589779322181708120639719, 6.16289593776083308319222379998, 7.17307885297614968893280201954, 7.81206313032829605943361950093, 8.602238465302687272981246570171
