| L(s) = 1 | + 3.82·5-s + 7-s + 5.86·11-s − 0.605·13-s − 2.04·17-s + 19-s + 5.86·23-s + 9.60·25-s − 2.04·29-s − 6.60·31-s + 3.82·35-s + 2·37-s + 5.86·41-s − 6.60·43-s + 2.04·47-s + 49-s + 3.82·53-s + 22.4·55-s + 11.7·59-s + 2·61-s − 2.31·65-s − 9.21·67-s − 15.5·71-s + 7.21·73-s + 5.86·77-s − 4·79-s − 3.82·83-s + ⋯ |
| L(s) = 1 | + 1.70·5-s + 0.377·7-s + 1.76·11-s − 0.167·13-s − 0.496·17-s + 0.229·19-s + 1.22·23-s + 1.92·25-s − 0.379·29-s − 1.18·31-s + 0.645·35-s + 0.328·37-s + 0.916·41-s − 1.00·43-s + 0.298·47-s + 0.142·49-s + 0.524·53-s + 3.02·55-s + 1.52·59-s + 0.256·61-s − 0.287·65-s − 1.12·67-s − 1.84·71-s + 0.843·73-s + 0.668·77-s − 0.450·79-s − 0.419·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.409825661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.409825661\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 + 3.21T + 97T^{2} \) |
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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
−8.635022662518162387235942433003, −7.28410495483941561353502992561, −6.82128666934031340362514529596, −6.05422123082313123851063337317, −5.49020893736721232574616467332, −4.67589074170689324771630098510, −3.78260056896974884122769912922, −2.69730918150489330677421750664, −1.79845129414368275112248748901, −1.14812882339191650853659708606,
1.14812882339191650853659708606, 1.79845129414368275112248748901, 2.69730918150489330677421750664, 3.78260056896974884122769912922, 4.67589074170689324771630098510, 5.49020893736721232574616467332, 6.05422123082313123851063337317, 6.82128666934031340362514529596, 7.28410495483941561353502992561, 8.635022662518162387235942433003
