| L(s) = 1 | + (−0.732 − 2.73i)2-s + (0.515 − 0.297i)3-s + (−6.92 + 4i)4-s + (−2.76 + 10.3i)5-s + (−1.18 − 1.18i)6-s + (−5.68 − 9.85i)7-s + (16 + 15.9i)8-s + (−40.3 + 69.8i)9-s + 30.2·10-s + 107. i·11-s + (−2.37 + 4.12i)12-s + (−29.7 + 111. i)13-s + (−22.7 + 22.7i)14-s + (1.64 + 6.14i)15-s + (31.9 − 55.4i)16-s + (479. − 128. i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 − 0.683i)2-s + (0.0572 − 0.0330i)3-s + (−0.433 + 0.250i)4-s + (−0.110 + 0.413i)5-s + (−0.0330 − 0.0330i)6-s + (−0.116 − 0.201i)7-s + (0.250 + 0.249i)8-s + (−0.497 + 0.862i)9-s + 0.302·10-s + 0.890i·11-s + (−0.0165 + 0.0286i)12-s + (−0.176 + 0.657i)13-s + (−0.116 + 0.116i)14-s + (0.00732 + 0.0273i)15-s + (0.124 − 0.216i)16-s + (1.65 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.870759 + 0.526510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.870759 + 0.526510i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 + 2.73i)T \) |
| 37 | \( 1 + (1.03e3 + 895. i)T \) |
| good | 3 | \( 1 + (-0.515 + 0.297i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (2.76 - 10.3i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (5.68 + 9.85i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 107. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (29.7 - 111. i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-479. + 128. i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (154. - 577. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (351. + 351. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (1.01e3 - 1.01e3i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-329. + 329. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.43e3 - 830. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.49e3 + 1.49e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 518.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (671. - 1.16e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (808. - 216. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-3.56e3 - 956. i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-6.77e3 + 3.91e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-3.09e3 - 5.36e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 7.59e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.11e3 + 4.14e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-1.50e3 + 2.60e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.31e3 - 8.63e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-4.58e3 - 4.58e3i)T + 8.85e7iT^{2} \) |
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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
−14.11162934160828384571134131745, −12.69559079370391340730900630678, −11.82867420335529178700994746551, −10.58232900111028322190121277003, −9.811106069826067373447481021242, −8.294916153709699838369699495902, −7.16914970495252132618128526658, −5.27380453858060405463160658409, −3.60233347494904899785039697727, −1.93032027100926895114355698083,
0.56283737098943479394860529112, 3.39025352059779764663273221032, 5.27920435428760117662285188580, 6.35887860592919299171170027688, 7.948752807713404692819029415218, 8.859808619663346373675466194288, 9.990069605491635313081603152957, 11.50766759719724384960575798193, 12.65083167629791018336951717431, 13.79516899312178558658301583775
