| L(s) = 1 | + (0.679 + 0.809i)2-s + (−4.59 − 3.85i)3-s + (1.19 − 6.77i)4-s + (−2.08 − 5.73i)5-s − 6.33i·6-s + (1.40 − 0.512i)7-s + (13.6 − 7.86i)8-s + (1.55 + 8.79i)9-s + (3.22 − 5.58i)10-s + (12.7 + 22.1i)11-s + (−31.6 + 26.5i)12-s + (4.95 + 0.874i)13-s + (1.37 + 0.791i)14-s + (−12.5 + 34.3i)15-s + (−36.1 − 13.1i)16-s + (28.6 − 5.04i)17-s + ⋯ |
| L(s) = 1 | + (0.240 + 0.286i)2-s + (−0.883 − 0.741i)3-s + (0.149 − 0.847i)4-s + (−0.186 − 0.513i)5-s − 0.431i·6-s + (0.0759 − 0.0276i)7-s + (0.602 − 0.347i)8-s + (0.0574 + 0.325i)9-s + (0.102 − 0.176i)10-s + (0.350 + 0.606i)11-s + (−0.760 + 0.637i)12-s + (0.105 + 0.0186i)13-s + (0.0261 + 0.0151i)14-s + (−0.215 + 0.591i)15-s + (−0.564 − 0.205i)16-s + (0.408 − 0.0720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.804198 - 0.739157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804198 - 0.739157i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (179. + 135. i)T \) |
| good | 2 | \( 1 + (-0.679 - 0.809i)T + (-1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (4.59 + 3.85i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (2.08 + 5.73i)T + (-95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-1.40 + 0.512i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-12.7 - 22.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-4.95 - 0.874i)T + (2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-28.6 + 5.04i)T + (4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-9.07 + 10.8i)T + (-1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-146. - 84.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-106. + 61.3i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 109. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-37.5 + 212. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 - 434. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (202. - 350. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-568. - 206. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-4.61 + 12.6i)T + (-1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (837. + 147. i)T + (2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (406. - 147. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-556. - 466. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (201. + 553. i)T + (-3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (42.9 + 243. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-158. + 436. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-834. - 482. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.62929343704379599632934308367, −14.47958794293562999561289510260, −13.11944496941146115556762269989, −12.04976980117216902550846111748, −10.91949806053967171328809796466, −9.370116438923361339387194653276, −7.32776491546525940662728009027, −6.16479379007997608778508216256, −4.86544944705208691663540752155, −1.10544934328575999519913366297,
3.35418702670395629039562950991, 4.99264640090728251699008345529, 6.84580117892205142506881433444, 8.548999520295742017866398059282, 10.47605056602575205222129121448, 11.26358116022276255058775537062, 12.25900190651600695615017777750, 13.70792037057147240340985714181, 15.16124722407505248842511343496, 16.48067997350669829983648587690
