| L(s) = 1 | + (55 − 10i)5-s + 4i·7-s − 500·11-s + 288i·13-s − 1.51e3i·17-s − 1.34e3·19-s + 4.10e3i·23-s + (2.92e3 − 1.10e3i)25-s − 2.64e3·29-s + 5.61e3·31-s + (40 + 220i)35-s − 7.28e3i·37-s + 1.89e4·41-s − 2.40e3i·43-s + 8.90e3i·47-s + ⋯ |
| L(s) = 1 | + (0.983 − 0.178i)5-s + 0.0308i·7-s − 1.24·11-s + 0.472i·13-s − 1.27i·17-s − 0.854·19-s + 1.61i·23-s + (0.936 − 0.352i)25-s − 0.584·29-s + 1.04·31-s + (0.00551 + 0.0303i)35-s − 0.875i·37-s + 1.76·41-s − 0.198i·43-s + 0.587i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.321781219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321781219\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-55 + 10i)T \) |
| good | 7 | \( 1 - 4iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 500T + 1.61e5T^{2} \) |
| 13 | \( 1 - 288iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.51e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.28e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.40e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.90e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.98e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.08e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.69e4iT - 8.58e9T^{2} \) |
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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
−9.540458870487694395300064259884, −9.078077093933950672159320907466, −7.87464266792861300427478796728, −7.10909930857722787393161259337, −5.94637505827390992288057143877, −5.31889058284927699686812506812, −4.32608676442173408033883569061, −2.83647755418166033097872403944, −2.07900925256954359702997241864, −0.75398859597034106508219486324,
0.64532435682918592109578917605, 2.07654885080261927482334681499, 2.77026127161829561322939346156, 4.18383958594029829078765163134, 5.26564638364668762197672391066, 6.05489603642641213182137360567, 6.83889315816418784470008467093, 8.107569897613492255549136308167, 8.609137764800264835456201115155, 9.870336688405155462033196033545
