| L(s) = 1 | + (−1.98 − 7.74i)2-s + (−56.1 + 30.7i)4-s − 106. i·5-s − 614. i·7-s + (350. + 373. i)8-s + (−826. + 212. i)10-s + 1.31e3·11-s − 626. i·13-s + (−4.75e3 + 1.22e3i)14-s + (2.19e3 − 3.45e3i)16-s − 8.17e3·17-s − 9.29e3·19-s + (3.28e3 + 5.98e3i)20-s + (−2.62e3 − 1.02e4i)22-s + 1.42e4i·23-s + ⋯ |
| L(s) = 1 | + (−0.248 − 0.968i)2-s + (−0.876 + 0.481i)4-s − 0.853i·5-s − 1.79i·7-s + (0.683 + 0.729i)8-s + (−0.826 + 0.212i)10-s + 0.991·11-s − 0.285i·13-s + (−1.73 + 0.444i)14-s + (0.536 − 0.843i)16-s − 1.66·17-s − 1.35·19-s + (0.410 + 0.748i)20-s + (−0.246 − 0.959i)22-s + 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.325797 + 0.751968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325797 + 0.751968i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.98 + 7.74i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 106. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 614. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.31e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 626. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 8.17e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.29e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.42e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.01e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.01e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.00e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.21e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 6.85e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 3.55e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.87e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.58e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.63e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.87e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.38e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.91e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.30e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 4.12e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.13e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 8.67e5T + 8.32e11T^{2} \) |
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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
−12.88107604860472292837564773172, −11.41463040752642137947215020287, −10.60150445539408128541243969724, −9.391706703074464197103895859802, −8.369251320294072359931942646587, −6.91483553773160601682807925554, −4.64051205627021355660710496039, −3.81078404242138612582019538453, −1.56791253002348638644449281110, −0.35300926439833876636645178971,
2.30962147319889227586860061414, 4.45415406834871158284683431373, 6.13921637841873262845965574929, 6.71485384201543436836771696017, 8.563695242444178890826338193312, 9.101201175022736190297090425098, 10.63532240224109775076162367821, 11.93143602816251028686730115281, 13.20611048589318320730982112260, 14.68864731442224357007243681360
