| L(s) = 1 | + 3.73·5-s + 7-s + 4.19·11-s + 0.464·13-s + 7·17-s + 2.73·19-s − 6.19·23-s + 8.92·25-s + 8.46·29-s + 2.19·31-s + 3.73·35-s − 6.66·37-s + 9.46·41-s − 5.46·43-s − 1.26·47-s + 49-s − 2.53·53-s + 15.6·55-s + 6.19·59-s − 9.92·61-s + 1.73·65-s + 3.26·67-s − 13.4·71-s + 11.7·73-s + 4.19·77-s − 15.1·79-s + 14.5·83-s + ⋯ |
| L(s) = 1 | + 1.66·5-s + 0.377·7-s + 1.26·11-s + 0.128·13-s + 1.69·17-s + 0.626·19-s − 1.29·23-s + 1.78·25-s + 1.57·29-s + 0.394·31-s + 0.630·35-s − 1.09·37-s + 1.47·41-s − 0.833·43-s − 0.184·47-s + 0.142·49-s − 0.348·53-s + 2.11·55-s + 0.806·59-s − 1.27·61-s + 0.214·65-s + 0.399·67-s − 1.59·71-s + 1.37·73-s + 0.478·77-s − 1.70·79-s + 1.60·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.017871430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.017871430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 + 9.92T + 61T^{2} \) |
| 67 | \( 1 - 3.26T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−7.79670471846539966980852319421, −6.84610853893905308120756336652, −6.28962580188393551898813250440, −5.72001575579244716508895722514, −5.16073619953963745087053981617, −4.27382700624429322936725427631, −3.36909815463941688253931505159, −2.55027313042668356488721491412, −1.54486882935590393829351554989, −1.13769389918677358038379142681,
1.13769389918677358038379142681, 1.54486882935590393829351554989, 2.55027313042668356488721491412, 3.36909815463941688253931505159, 4.27382700624429322936725427631, 5.16073619953963745087053981617, 5.72001575579244716508895722514, 6.28962580188393551898813250440, 6.84610853893905308120756336652, 7.79670471846539966980852319421
