| L(s) = 1 | + (2.06 + 4.76i)3-s + (−0.845 + 1.46i)5-s + (8.57 + 14.8i)7-s + (−18.4 + 19.6i)9-s + (2.00 + 3.46i)11-s + (−20.5 + 35.5i)13-s + (−8.73 − 1.01i)15-s + 3.32·17-s + 108.·19-s + (−53.1 + 71.5i)21-s + (71.2 − 123. i)23-s + (61.0 + 105. i)25-s + (−131. − 47.5i)27-s + (−147. − 255. i)29-s + (119. − 207. i)31-s + ⋯ |
| L(s) = 1 | + (0.397 + 0.917i)3-s + (−0.0756 + 0.131i)5-s + (0.462 + 0.801i)7-s + (−0.684 + 0.728i)9-s + (0.0548 + 0.0950i)11-s + (−0.437 + 0.758i)13-s + (−0.150 − 0.0174i)15-s + 0.0474·17-s + 1.31·19-s + (−0.551 + 0.742i)21-s + (0.646 − 1.11i)23-s + (0.488 + 0.846i)25-s + (−0.940 − 0.339i)27-s + (−0.945 − 1.63i)29-s + (0.694 − 1.20i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.16567 + 1.10207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16567 + 1.10207i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.06 - 4.76i)T \) |
| good | 5 | \( 1 + (0.845 - 1.46i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-8.57 - 14.8i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-2.00 - 3.46i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (20.5 - 35.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 3.32T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-71.2 + 123. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (147. + 255. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-119. + 207. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-172. + 298. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-210. - 364. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-46.3 - 80.3i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 191.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (330. - 573. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-179. - 311. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-136. + 237. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 824.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (144. + 250. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-660. - 1.14e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 328.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (383. + 665. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.65419596462104444736326911299, −13.55752532826845514433965418581, −11.96574026776740713787634977087, −11.09911368326401897103492738810, −9.707376396754631477786723705572, −8.892917072359013757896808783553, −7.55762349115004940035146118605, −5.64137085628706943937886508824, −4.34586696355571949550042665540, −2.57855668804133302832329631868,
1.13642032917802026799840792225, 3.24337356508585307706516949206, 5.27087253509351615020337036914, 7.03191735685064164404601691501, 7.82145315547262417123412628323, 9.108717764314196123175097950072, 10.60063042516907329423971689261, 11.84711405291217783562021943945, 12.86744558029065734756798907306, 13.88138815126605629925634632014
