| L(s) = 1 | + (−2.93 + 0.787i)2-s + (4.37 − 2.52i)3-s + (−5.83 + 3.36i)4-s + (10.5 + 2.82i)5-s + (−10.8 + 10.8i)6-s + (28.0 + 48.5i)7-s + (48.9 − 48.9i)8-s + (−27.7 + 48.0i)9-s − 33.2·10-s + 201. i·11-s + (−17.0 + 29.4i)12-s + (191. + 51.2i)13-s + (−120. − 120. i)14-s + (53.2 − 14.2i)15-s + (−51.3 + 89.0i)16-s + (−77.4 − 288. i)17-s + ⋯ |
| L(s) = 1 | + (−0.734 + 0.196i)2-s + (0.486 − 0.280i)3-s + (−0.364 + 0.210i)4-s + (0.421 + 0.113i)5-s + (−0.302 + 0.302i)6-s + (0.572 + 0.991i)7-s + (0.764 − 0.764i)8-s + (−0.342 + 0.592i)9-s − 0.332·10-s + 1.66i·11-s + (−0.118 + 0.204i)12-s + (1.13 + 0.303i)13-s + (−0.615 − 0.615i)14-s + (0.236 − 0.0634i)15-s + (−0.200 + 0.347i)16-s + (−0.267 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.901771 + 0.626958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.901771 + 0.626958i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.06e3 + 863. i)T \) |
| good | 2 | \( 1 + (2.93 - 0.787i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-4.37 + 2.52i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-10.5 - 2.82i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (-28.0 - 48.5i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 201. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-191. - 51.2i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (77.4 + 288. i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (338. + 90.5i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-548. + 548. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-1.01e3 - 1.01e3i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (381. + 381. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (485. - 280. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (59.3 - 59.3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 3.24e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-930. + 1.61e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (631. + 2.35e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (291. - 1.08e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-2.75e3 + 1.59e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-570. - 988. i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 3.68e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (287. + 77.0i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-2.59e3 + 4.50e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.95e3 + 1.59e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (6.51e3 - 6.51e3i)T - 8.85e7iT^{2} \) |
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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
−15.95639781993156077336551387427, −14.65470564683649758593675353112, −13.52939629525074206926781856152, −12.38812658378983625108291966712, −10.66871535630565680080320604178, −9.144862858106275904476673810767, −8.437018348804152097956654745457, −7.00146070721691035885425371337, −4.81828852485561394669783864427, −2.12741405113541389944403761172,
1.03609513408325795158114907993, 3.82683990301554829416503904165, 5.90633032924011626408584419071, 8.237184203429217273892763859682, 8.885464058815545871051072606176, 10.37827237538846075934690309645, 11.17793656624218133906861080433, 13.52001761515721152607147122371, 13.94706389331357547116628042167, 15.36196518886233200748732319342
