| L(s) = 1 | − 25·5-s − 94.1·7-s + 143.·11-s + 421.·13-s − 1.98e3·17-s + 1.31e3·19-s − 4.02e3·23-s + 625·25-s + 6.41e3·29-s + 2.35e3·31-s + 2.35e3·35-s − 7.87e3·37-s − 1.50e4·41-s − 1.14e3·43-s − 2.15e4·47-s − 7.94e3·49-s − 9.56e3·53-s − 3.59e3·55-s + 4.27e4·59-s + 3.21e4·61-s − 1.05e4·65-s + 3.03e4·67-s + 3.60e4·71-s − 6.34e4·73-s − 1.35e4·77-s + 8.99e4·79-s + 3.82e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.726·7-s + 0.358·11-s + 0.691·13-s − 1.66·17-s + 0.837·19-s − 1.58·23-s + 0.200·25-s + 1.41·29-s + 0.439·31-s + 0.324·35-s − 0.945·37-s − 1.40·41-s − 0.0941·43-s − 1.42·47-s − 0.472·49-s − 0.467·53-s − 0.160·55-s + 1.59·59-s + 1.10·61-s − 0.309·65-s + 0.826·67-s + 0.847·71-s − 1.39·73-s − 0.260·77-s + 1.62·79-s + 0.608·83-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
| \(L(3)\) |
\(\approx\) |
\(1.324777943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324777943\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 94.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.74e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.58e9T^{2} \) |
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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
−9.704719138391626311912566461360, −8.685152811573365968711515932327, −8.098964135258472102089649414937, −6.74127870402496888699261072675, −6.42853755015557943708057951868, −5.07885429493266062411514021931, −4.03139355990114457127833519159, −3.21705653086219638303971498036, −1.92191569206915333981724482023, −0.52536267852425195496422192860,
0.52536267852425195496422192860, 1.92191569206915333981724482023, 3.21705653086219638303971498036, 4.03139355990114457127833519159, 5.07885429493266062411514021931, 6.42853755015557943708057951868, 6.74127870402496888699261072675, 8.098964135258472102089649414937, 8.685152811573365968711515932327, 9.704719138391626311912566461360
