| L(s) = 1 | + (−1.05 − 2.90i)2-s + (4.78 + 1.74i)3-s + (−1.21 + 1.01i)4-s + (7.19 − 1.26i)5-s − 15.7i·6-s + (−0.149 − 0.848i)7-s + (−17.2 − 9.93i)8-s + (−0.843 − 0.708i)9-s + (−11.3 − 19.5i)10-s + (−0.165 + 0.285i)11-s + (−7.56 + 2.75i)12-s + (35.7 + 42.5i)13-s + (−2.31 + 1.33i)14-s + (36.6 + 6.45i)15-s + (−12.8 + 73.0i)16-s + (−23.8 + 28.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.374 − 1.02i)2-s + (0.920 + 0.334i)3-s + (−0.151 + 0.127i)4-s + (0.643 − 0.113i)5-s − 1.07i·6-s + (−0.00808 − 0.0458i)7-s + (−0.760 − 0.439i)8-s + (−0.0312 − 0.0262i)9-s + (−0.357 − 0.619i)10-s + (−0.00452 + 0.00783i)11-s + (−0.181 + 0.0662i)12-s + (0.762 + 0.908i)13-s + (−0.0441 + 0.0254i)14-s + (0.630 + 0.111i)15-s + (−0.201 + 1.14i)16-s + (−0.340 + 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20651 - 0.841612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20651 - 0.841612i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (142. - 174. i)T \) |
| good | 2 | \( 1 + (1.05 + 2.90i)T + (-6.12 + 5.14i)T^{2} \) |
| 3 | \( 1 + (-4.78 - 1.74i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-7.19 + 1.26i)T + (117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (0.149 + 0.848i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (0.165 - 0.285i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-35.7 - 42.5i)T + (-381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (23.8 - 28.4i)T + (-853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (32.7 - 90.0i)T + (-5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-75.6 + 43.6i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (12.2 + 7.08i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 7.96iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-226. + 189. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 192. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (305. + 529. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (18.8 - 107. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-380. - 67.1i)T + (1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (287. + 342. i)T + (-3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (154. + 876. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (257. + 93.8i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 - 883.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (161. - 28.5i)T + (4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (1.02e3 + 863. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-527. - 93.0i)T + (6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-678. + 391. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.48364241075224580339813920150, −14.39486624108855869498831084610, −13.25393617425901743131092829285, −11.83736215156949864232801425319, −10.54048480297709988078227191015, −9.454992655433424026199665795472, −8.579802985291691271725816860494, −6.25296207179421097395972527739, −3.66597770109953557167360436115, −1.95924195331661620640422368371,
2.73511942225474823284143024813, 5.70753932573639049525840730626, 7.14618244905355532839851220396, 8.334086160467908950488142677841, 9.270936926605092030827878606402, 11.11561409271462615603569386572, 13.00378489699668553746800859404, 13.96844253288823192291882898356, 15.05832228024084213311760743416, 15.98064352069921811687178816457
