| L(s) = 1 | + 25·5-s − 15.4·7-s + 492.·11-s + 678.·13-s + 910.·17-s + 1.71e3·19-s − 2.42e3·23-s + 625·25-s + 1.75e3·29-s − 682.·31-s − 385.·35-s + 7.97e3·37-s − 1.53e4·41-s − 1.41e4·43-s + 5.67e3·47-s − 1.65e4·49-s + 1.64e4·53-s + 1.23e4·55-s + 1.93e4·59-s + 4.93e4·61-s + 1.69e4·65-s − 2.41e4·67-s − 4.04e4·71-s + 4.66e4·73-s − 7.58e3·77-s − 8.15e4·79-s + 8.01e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.118·7-s + 1.22·11-s + 1.11·13-s + 0.764·17-s + 1.08·19-s − 0.954·23-s + 0.200·25-s + 0.387·29-s − 0.127·31-s − 0.0531·35-s + 0.957·37-s − 1.42·41-s − 1.16·43-s + 0.374·47-s − 0.985·49-s + 0.802·53-s + 0.548·55-s + 0.725·59-s + 1.69·61-s + 0.497·65-s − 0.656·67-s − 0.951·71-s + 1.02·73-s − 0.145·77-s − 1.47·79-s + 1.27·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.074630361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074630361\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 + 15.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 492.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 678.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 910.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 682.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.53e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.64e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.32e3T + 8.58e9T^{2} \) |
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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
−9.710348622273280132995769050980, −8.831604919069155924884610155206, −8.024373448485025825420182972253, −6.86915252162275759109965716728, −6.14593790011883915014112945618, −5.28889388126237706173571457379, −3.99362103740267691143574227208, −3.21141805315634369140671641643, −1.73378010306059977302117641320, −0.883408441473889663538316932514,
0.883408441473889663538316932514, 1.73378010306059977302117641320, 3.21141805315634369140671641643, 3.99362103740267691143574227208, 5.28889388126237706173571457379, 6.14593790011883915014112945618, 6.86915252162275759109965716728, 8.024373448485025825420182972253, 8.831604919069155924884610155206, 9.710348622273280132995769050980
