| L(s) = 1 | + (0.236 − 0.649i)2-s + (−6.38 + 2.32i)3-s + (5.76 + 4.83i)4-s + (12.1 + 2.13i)5-s + 4.69i·6-s + (−5.32 + 30.1i)7-s + (9.29 − 5.36i)8-s + (14.6 − 12.3i)9-s + (4.25 − 7.37i)10-s + (−22.3 − 38.7i)11-s + (−48.0 − 17.4i)12-s + (26.0 − 31.0i)13-s + (18.3 + 10.6i)14-s + (−82.3 + 14.5i)15-s + (9.16 + 51.9i)16-s + (43.0 + 51.2i)17-s + ⋯ |
| L(s) = 1 | + (0.0836 − 0.229i)2-s + (−1.22 + 0.447i)3-s + (0.720 + 0.604i)4-s + (1.08 + 0.191i)5-s + 0.319i·6-s + (−0.287 + 1.63i)7-s + (0.410 − 0.237i)8-s + (0.543 − 0.456i)9-s + (0.134 − 0.233i)10-s + (−0.613 − 1.06i)11-s + (−1.15 − 0.420i)12-s + (0.556 − 0.662i)13-s + (0.350 + 0.202i)14-s + (−1.41 + 0.250i)15-s + (0.143 + 0.811i)16-s + (0.614 + 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.03880 + 0.562415i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03880 + 0.562415i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (100. + 201. i)T \) |
| good | 2 | \( 1 + (-0.236 + 0.649i)T + (-6.12 - 5.14i)T^{2} \) |
| 3 | \( 1 + (6.38 - 2.32i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (-12.1 - 2.13i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (5.32 - 30.1i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (22.3 + 38.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-26.0 + 31.0i)T + (-381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-43.0 - 51.2i)T + (-853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (29.9 + 82.2i)T + (-5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-40.7 - 23.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-170. + 98.4i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 36.9iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (167. + 140. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 - 453. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-223. + 386. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-92.2 - 523. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-60.5 + 10.6i)T + (1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-139. + 166. i)T + (-3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (27.0 - 153. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (337. - 122. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + 853.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-304. - 53.6i)T + (4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (90.6 - 76.0i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-475. + 83.8i)T + (6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (963. + 556. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.09134543351525608685897529595, −15.37995223437228534104657641588, −13.34245311352746004847016311465, −12.26696153230165317226195294239, −11.19990902313124699758510270585, −10.28539301121178109576726436314, −8.548543102252133594088367514951, −6.17780110310318570978828120810, −5.59968808215914238006150752520, −2.72980199551245161225917612198,
1.34114672243812208638952942679, 5.07097740196995023283162976394, 6.39298252367233913558818235190, 7.20174477028451833546768888942, 10.03830690786600844705554728323, 10.59586311698276540565204661958, 12.03089911156449556515084374228, 13.37592233308432646749621265615, 14.34523279938016926271903695688, 16.14880217159320993798099135403
