| L(s) = 1 | − 2.64·5-s + 7-s − 0.645·11-s + 5.29·13-s + 19-s + 1.35·23-s + 2.00·25-s − 6·29-s − 2.29·31-s − 2.64·35-s − 2.29·37-s − 8.64·41-s + 7.29·43-s + 11.2·47-s + 49-s + 1.70·55-s − 3.29·59-s − 6.58·61-s − 14.0·65-s + 1.29·67-s + 13.2·71-s + 0.708·73-s − 0.645·77-s + 11.2·79-s + 10·83-s + 13.9·89-s + 5.29·91-s + ⋯ |
| L(s) = 1 | − 1.18·5-s + 0.377·7-s − 0.194·11-s + 1.46·13-s + 0.229·19-s + 0.282·23-s + 0.400·25-s − 1.11·29-s − 0.411·31-s − 0.447·35-s − 0.376·37-s − 1.35·41-s + 1.11·43-s + 1.64·47-s + 0.142·49-s + 0.230·55-s − 0.428·59-s − 0.842·61-s − 1.73·65-s + 0.157·67-s + 1.56·71-s + 0.0829·73-s − 0.0735·77-s + 1.27·79-s + 1.09·83-s + 1.47·89-s + 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472158636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472158636\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.29T + 37T^{2} \) |
| 41 | \( 1 + 8.64T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 1.29T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.729119403403450663487493329435, −7.83672587032887635920019989122, −7.49109259480459555614805427469, −6.49175830333650530008332078977, −5.64992863100969220083769518933, −4.78179414320327753104482914099, −3.81421960798734821151372032567, −3.42148458654315459055090734582, −2.00287651010499383647707168627, −0.74807549459243729831945013422,
0.74807549459243729831945013422, 2.00287651010499383647707168627, 3.42148458654315459055090734582, 3.81421960798734821151372032567, 4.78179414320327753104482914099, 5.64992863100969220083769518933, 6.49175830333650530008332078977, 7.49109259480459555614805427469, 7.83672587032887635920019989122, 8.729119403403450663487493329435
