| L(s) = 1 | − 0.755·3-s − 25·5-s − 172.·7-s − 242.·9-s + 391.·11-s + 149.·13-s + 18.8·15-s + 1.18e3·17-s + 685.·19-s + 130.·21-s + 996.·23-s + 625·25-s + 366.·27-s − 8.76e3·29-s + 9.52e3·31-s − 295.·33-s + 4.31e3·35-s + 1.02e4·37-s − 112.·39-s + 32.6·41-s + 1.03e4·43-s + 6.06e3·45-s + 1.69e4·47-s + 1.29e4·49-s − 897.·51-s − 2.22e4·53-s − 9.78e3·55-s + ⋯ |
| L(s) = 1 | − 0.0484·3-s − 0.447·5-s − 1.33·7-s − 0.997·9-s + 0.975·11-s + 0.244·13-s + 0.0216·15-s + 0.996·17-s + 0.435·19-s + 0.0644·21-s + 0.392·23-s + 0.200·25-s + 0.0968·27-s − 1.93·29-s + 1.78·31-s − 0.0472·33-s + 0.594·35-s + 1.22·37-s − 0.0118·39-s + 0.00303·41-s + 0.851·43-s + 0.446·45-s + 1.11·47-s + 0.768·49-s − 0.0483·51-s − 1.08·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.327926619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327926619\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 3 | \( 1 + 0.755T + 243T^{2} \) |
| 7 | \( 1 + 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 391.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 149.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 685.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 996.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 32.6T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e4T + 8.58e9T^{2} \) |
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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
−11.95317728366059039626853856055, −11.15303493580397217843247313350, −9.791577168237250152630951500107, −9.050740148067776304426882051682, −7.79860942012218930363787089575, −6.56074574522555306228522901662, −5.65216840625499503997222915924, −3.88593577394161215316757927966, −2.91831087837439195192114925289, −0.74244313692243571320409817133,
0.74244313692243571320409817133, 2.91831087837439195192114925289, 3.88593577394161215316757927966, 5.65216840625499503997222915924, 6.56074574522555306228522901662, 7.79860942012218930363787089575, 9.050740148067776304426882051682, 9.791577168237250152630951500107, 11.15303493580397217843247313350, 11.95317728366059039626853856055
