| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 1.73·5-s + 7-s − 1.73·8-s + 2.99·10-s − 1.73·11-s + 2·13-s + 1.73·14-s − 5·16-s − 6.92·17-s + 5·19-s + 1.73·20-s − 2.99·22-s + 1.73·23-s − 2.00·25-s + 3.46·26-s + 0.999·28-s − 10.3·29-s + 5·31-s − 5.19·32-s − 11.9·34-s + 1.73·35-s − 7·37-s + 8.66·38-s − 3.00·40-s + 5.19·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.774·5-s + 0.377·7-s − 0.612·8-s + 0.948·10-s − 0.522·11-s + 0.554·13-s + 0.462·14-s − 1.25·16-s − 1.68·17-s + 1.14·19-s + 0.387·20-s − 0.639·22-s + 0.361·23-s − 0.400·25-s + 0.679·26-s + 0.188·28-s − 1.92·29-s + 0.898·31-s − 0.918·32-s − 2.05·34-s + 0.292·35-s − 1.15·37-s + 1.40·38-s − 0.474·40-s + 0.811·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.116711015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116711015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−12.95347004378688994697617700247, −11.73041355137575571035658155128, −10.91639014766042853271969197795, −9.576964852801978258484979874207, −8.638982879276385147539891965623, −7.08197461498358252278819297929, −5.87699223908590498863469093575, −5.11031402764737521030468941467, −3.84755291172920983413472991962, −2.31763303042601749523987733960,
2.31763303042601749523987733960, 3.84755291172920983413472991962, 5.11031402764737521030468941467, 5.87699223908590498863469093575, 7.08197461498358252278819297929, 8.638982879276385147539891965623, 9.576964852801978258484979874207, 10.91639014766042853271969197795, 11.73041355137575571035658155128, 12.95347004378688994697617700247
