| L(s) = 1 | + 3.61·5-s + 7-s + 11-s + 5.09·13-s + 5·17-s − 3.47·19-s − 23-s + 8.09·25-s + 6.23·29-s − 8.70·31-s + 3.61·35-s + 1.47·37-s + 5.76·41-s + 6.32·43-s − 3.70·47-s + 49-s − 0.381·53-s + 3.61·55-s + 3.61·59-s − 7.56·61-s + 18.4·65-s − 7.32·67-s + 4.32·71-s − 0.527·73-s + 77-s − 10.7·79-s − 9.18·83-s + ⋯ |
| L(s) = 1 | + 1.61·5-s + 0.377·7-s + 0.301·11-s + 1.41·13-s + 1.21·17-s − 0.796·19-s − 0.208·23-s + 1.61·25-s + 1.15·29-s − 1.56·31-s + 0.611·35-s + 0.242·37-s + 0.900·41-s + 0.964·43-s − 0.540·47-s + 0.142·49-s − 0.0524·53-s + 0.487·55-s + 0.471·59-s − 0.968·61-s + 2.28·65-s − 0.895·67-s + 0.513·71-s − 0.0617·73-s + 0.113·77-s − 1.20·79-s − 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.467686942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.467686942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.381T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 + 7.32T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 + 0.527T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.232611221598618804420622397373, −7.37563827772298981615523056929, −6.43949899567708053277612264812, −5.94816560563640021296464873658, −5.48979915121249253825492710754, −4.52055330979422112952973650888, −3.63172483311619236650745555410, −2.67534111392347847446612059125, −1.73574360971961327421751698224, −1.10807358520410350388947720363,
1.10807358520410350388947720363, 1.73574360971961327421751698224, 2.67534111392347847446612059125, 3.63172483311619236650745555410, 4.52055330979422112952973650888, 5.48979915121249253825492710754, 5.94816560563640021296464873658, 6.43949899567708053277612264812, 7.37563827772298981615523056929, 8.232611221598618804420622397373
