| L(s) = 1 | + (−0.746 − 0.889i)2-s + (3.53 + 2.96i)3-s + (1.15 − 6.55i)4-s + (3.46 + 9.50i)5-s − 5.35i·6-s + (25.4 − 9.24i)7-s + (−14.7 + 8.50i)8-s + (−0.996 − 5.65i)9-s + (5.87 − 10.1i)10-s + (13.3 + 23.0i)11-s + (23.4 − 19.7i)12-s + (−46.5 − 8.20i)13-s + (−27.1 − 15.6i)14-s + (−15.9 + 43.8i)15-s + (−31.4 − 11.4i)16-s + (−73.6 + 12.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.263 − 0.314i)2-s + (0.679 + 0.570i)3-s + (0.144 − 0.818i)4-s + (0.309 + 0.850i)5-s − 0.364i·6-s + (1.37 − 0.499i)7-s + (−0.651 + 0.375i)8-s + (−0.0369 − 0.209i)9-s + (0.185 − 0.321i)10-s + (0.364 + 0.632i)11-s + (0.565 − 0.474i)12-s + (−0.993 − 0.175i)13-s + (−0.519 − 0.299i)14-s + (−0.274 + 0.754i)15-s + (−0.491 − 0.178i)16-s + (−1.05 + 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.46985 - 0.142476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46985 - 0.142476i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (179. - 136. i)T \) |
| good | 2 | \( 1 + (0.746 + 0.889i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (-3.53 - 2.96i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (-3.46 - 9.50i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-25.4 + 9.24i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-13.3 - 23.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (46.5 + 8.20i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (73.6 - 12.9i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (73.1 - 87.1i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (53.1 + 30.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-169. + 98.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 31.0iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (56.8 - 322. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + 376. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (116. - 201. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-99.4 - 36.1i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-69.8 + 191. i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-787. - 138. i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-96.9 + 35.2i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (529. + 444. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (166. + 457. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-254. - 1.44e3i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-270. + 742. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-139. - 80.4i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.29242417020838840867564247005, −14.64328419527777869038331618332, −14.14223392756992574308148722421, −11.87999798781212699203252222723, −10.56658734091728009372209121359, −9.909460891593036922930426845132, −8.403284049019476454520991283724, −6.58713783770153752265248316003, −4.52487165207518832974288724255, −2.18920926571281448873427172766,
2.24504111493100756061445247678, 4.86246713672293147629946247884, 7.09056114855955753908421217552, 8.481280506231817529451463005677, 8.816227271175793010696457772532, 11.29375661427048396915037037879, 12.46296274880981955574152617322, 13.52642007315960868990523163039, 14.68395534425093278920551653723, 16.07176782607118434282113306079
