| L(s) = 1 | + 3.38·5-s − 7·7-s − 11.3·11-s + 7.52·13-s − 30.4·17-s + 108.·19-s + 185.·23-s − 113.·25-s + 120·29-s − 79.6·31-s − 23.6·35-s + 37.6·37-s + 202.·41-s + 45.9·43-s + 158.·47-s + 49·49-s + 405.·53-s − 38.4·55-s + 625.·59-s − 567.·61-s + 25.4·65-s + 415.·67-s + 148.·71-s − 86.9·73-s + 79.6·77-s + 807.·79-s + 401.·83-s + ⋯ |
| L(s) = 1 | + 0.302·5-s − 0.377·7-s − 0.311·11-s + 0.160·13-s − 0.434·17-s + 1.31·19-s + 1.68·23-s − 0.908·25-s + 0.768·29-s − 0.461·31-s − 0.114·35-s + 0.167·37-s + 0.772·41-s + 0.162·43-s + 0.490·47-s + 0.142·49-s + 1.05·53-s − 0.0943·55-s + 1.38·59-s − 1.19·61-s + 0.0485·65-s + 0.756·67-s + 0.247·71-s − 0.139·73-s + 0.117·77-s + 1.15·79-s + 0.530·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.965803911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965803911\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 7T \) |
| good | 5 | \( 1 - 3.38T + 125T^{2} \) |
| 11 | \( 1 + 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.52T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 202.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 45.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 405.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 625.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 567.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 415.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 148.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 86.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 807.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 401.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 246.T + 9.12e5T^{2} \) |
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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
−10.49304716469533656214636530679, −9.564219932458285574263598837246, −8.894557704769217434643863459032, −7.71724263722004002214227235346, −6.87630081675754048872608262211, −5.81596277979424217162545164301, −4.91356137399442951219418006631, −3.57331242906815176909015298728, −2.46272982240083895264139153869, −0.886277541122276962470229507742,
0.886277541122276962470229507742, 2.46272982240083895264139153869, 3.57331242906815176909015298728, 4.91356137399442951219418006631, 5.81596277979424217162545164301, 6.87630081675754048872608262211, 7.71724263722004002214227235346, 8.894557704769217434643863459032, 9.564219932458285574263598837246, 10.49304716469533656214636530679
