| L(s) = 1 | − 23.5·3-s − 30.4·5-s + 310.·9-s + 356.·11-s − 761.·13-s + 715.·15-s + 502.·17-s − 1.73e3·19-s − 3.56e3·23-s − 2.20e3·25-s − 1.58e3·27-s + 4.52e3·29-s − 7.13e3·31-s − 8.39e3·33-s − 1.34e4·37-s + 1.79e4·39-s − 1.22e4·41-s + 5.42e3·43-s − 9.43e3·45-s + 3.87e3·47-s − 1.18e4·51-s + 6.89e3·53-s − 1.08e4·55-s + 4.07e4·57-s + 3.66e4·59-s − 4.53e4·61-s + 2.31e4·65-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 0.543·5-s + 1.27·9-s + 0.889·11-s − 1.24·13-s + 0.820·15-s + 0.421·17-s − 1.10·19-s − 1.40·23-s − 0.704·25-s − 0.419·27-s + 0.999·29-s − 1.33·31-s − 1.34·33-s − 1.61·37-s + 1.88·39-s − 1.13·41-s + 0.447·43-s − 0.694·45-s + 0.255·47-s − 0.636·51-s + 0.337·53-s − 0.483·55-s + 1.66·57-s + 1.36·59-s − 1.55·61-s + 0.679·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4309639891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4309639891\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 23.5T + 243T^{2} \) |
| 5 | \( 1 + 30.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 356.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 761.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 502.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.49e4T + 8.58e9T^{2} \) |
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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
−10.54241908866557797007109743547, −9.917727493340333960346261475388, −8.630625902419353932263308365847, −7.43304143140481099019772408764, −6.61007608115271528181006067807, −5.70468955949995298137035864045, −4.71748886728911938277685820779, −3.77900288364682692203188575838, −1.88482305685359456788372116489, −0.36775918390688411906987984754,
0.36775918390688411906987984754, 1.88482305685359456788372116489, 3.77900288364682692203188575838, 4.71748886728911938277685820779, 5.70468955949995298137035864045, 6.61007608115271528181006067807, 7.43304143140481099019772408764, 8.630625902419353932263308365847, 9.917727493340333960346261475388, 10.54241908866557797007109743547
