| L(s) = 1 | − 0.936·5-s + 7-s − 4.97·11-s + 1.24·13-s − 5.22·17-s + 5.18·19-s + 2.00·23-s − 4.12·25-s − 6.87·29-s − 5.73·31-s − 0.936·35-s + 9.73·37-s + 11.4·41-s + 9.60·43-s + 1.96·47-s + 49-s + 7.63·53-s + 4.65·55-s − 4.87·59-s − 3.04·61-s − 1.16·65-s − 1.14·67-s + 8.83·71-s + 6.10·73-s − 4.97·77-s − 12.1·79-s + 0.862·83-s + ⋯ |
| L(s) = 1 | − 0.418·5-s + 0.377·7-s − 1.49·11-s + 0.345·13-s − 1.26·17-s + 1.18·19-s + 0.418·23-s − 0.824·25-s − 1.27·29-s − 1.02·31-s − 0.158·35-s + 1.59·37-s + 1.79·41-s + 1.46·43-s + 0.287·47-s + 0.142·49-s + 1.04·53-s + 0.628·55-s − 0.635·59-s − 0.389·61-s − 0.144·65-s − 0.140·67-s + 1.04·71-s + 0.714·73-s − 0.566·77-s − 1.36·79-s + 0.0946·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410048563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410048563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.936T + 5T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 8.83T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 0.862T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 7.57T + 97T^{2} \) |
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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
−8.123878604325263073211646834095, −7.62294488757367311545008711313, −7.18040395097073981363381998984, −5.91512274541796578652451983202, −5.49893620776071665237511191232, −4.54713231719433730717316007497, −3.88147295657897978553189170768, −2.81144476722588910566486572743, −2.06689414400170080115379847026, −0.64389156369771366813049623263,
0.64389156369771366813049623263, 2.06689414400170080115379847026, 2.81144476722588910566486572743, 3.88147295657897978553189170768, 4.54713231719433730717316007497, 5.49893620776071665237511191232, 5.91512274541796578652451983202, 7.18040395097073981363381998984, 7.62294488757367311545008711313, 8.123878604325263073211646834095
