| L(s) = 1 | − 3-s − 0.189·7-s + 9-s + 4.33·11-s + 4.67·13-s + 0.0397·17-s + 7.19·19-s + 0.189·21-s + 23-s − 27-s + 6.49·29-s + 9.17·31-s − 4.33·33-s − 3.03·37-s − 4.67·39-s − 8.69·41-s + 7.83·43-s − 5.03·47-s − 6.96·49-s − 0.0397·51-s + 6.96·53-s − 7.19·57-s + 4.64·59-s + 7.86·61-s − 0.189·63-s − 2.52·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.0717·7-s + 0.333·9-s + 1.30·11-s + 1.29·13-s + 0.00963·17-s + 1.65·19-s + 0.0414·21-s + 0.208·23-s − 0.192·27-s + 1.20·29-s + 1.64·31-s − 0.754·33-s − 0.499·37-s − 0.747·39-s − 1.35·41-s + 1.19·43-s − 0.733·47-s − 0.994·49-s − 0.00556·51-s + 0.956·53-s − 0.953·57-s + 0.604·59-s + 1.00·61-s − 0.0239·63-s − 0.308·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.254351472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254351472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 0.189T + 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 0.0397T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 + 5.03T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 - 4.64T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 - 0.417T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 + 0.0584T + 97T^{2} \) |
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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
−8.089878411588221750272639561490, −6.90805536205564439811416066208, −6.68377380714386471819196524064, −5.88914304104926839959433568914, −5.20618236555793658399101681193, −4.35189049186832221049015227782, −3.64366358021857203108368812568, −2.87760821260636691746994239606, −1.43302628979713800623179021272, −0.920535612805249648490265654501,
0.920535612805249648490265654501, 1.43302628979713800623179021272, 2.87760821260636691746994239606, 3.64366358021857203108368812568, 4.35189049186832221049015227782, 5.20618236555793658399101681193, 5.88914304104926839959433568914, 6.68377380714386471819196524064, 6.90805536205564439811416066208, 8.089878411588221750272639561490
