| L(s) = 1 | + (−0.0603 + 0.342i)2-s + (−0.439 − 2.49i)3-s + (1.76 + 0.642i)4-s + (−2.87 + 2.41i)5-s + 0.879·6-s + (−0.0923 + 0.0775i)7-s + (−0.673 + 1.16i)8-s + (−3.20 + 1.16i)9-s + (−0.652 − 1.13i)10-s + (0.766 − 1.32i)11-s + (0.826 − 4.68i)12-s + (−4.14 − 1.50i)13-s + (−0.0209 − 0.0362i)14-s + (7.29 + 6.11i)15-s + (2.52 + 2.11i)16-s + (4.60 − 1.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.0426 + 0.241i)2-s + (−0.253 − 1.43i)3-s + (0.883 + 0.321i)4-s + (−1.28 + 1.08i)5-s + 0.359·6-s + (−0.0349 + 0.0293i)7-s + (−0.238 + 0.412i)8-s + (−1.06 + 0.388i)9-s + (−0.206 − 0.357i)10-s + (0.230 − 0.400i)11-s + (0.238 − 1.35i)12-s + (−1.14 − 0.418i)13-s + (−0.00559 − 0.00969i)14-s + (1.88 + 1.57i)15-s + (0.630 + 0.528i)16-s + (1.11 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.690065 - 0.0901155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.690065 - 0.0901155i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (3.29 - 5.11i)T \) |
| good | 2 | \( 1 + (0.0603 - 0.342i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.439 + 2.49i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (2.87 - 2.41i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.0923 - 0.0775i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.766 + 1.32i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.14 + 1.50i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.60 + 1.67i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.224 - 1.27i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (1.43 + 2.49i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.620 - 1.07i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 41 | \( 1 + (1.11 + 0.405i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 + (-2.84 - 4.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.92 - 3.29i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (4.53 + 3.80i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.21 + 2.62i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.28 - 3.59i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.42 + 8.08i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 + (-9.95 + 8.35i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.2 + 4.44i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.23 - 4.39i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.875 - 1.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.42692710581644376819871777360, −15.19943596842197104547525523663, −14.20131875246285829519417005886, −12.27042021192047641198946614964, −11.95972201016394140126776651031, −10.70057737970740244238321334138, −7.84643963426870028962146613494, −7.41279366559629578440224364799, −6.27976561353757255676991802664, −2.95137205696863067204983213851,
3.88954016223025004976072620825, 5.21912720449961584878678950762, 7.52678807041830997071637963055, 9.277775875291471844401526164526, 10.35494202585043987657811830596, 11.67109839830884235669957170865, 12.29886238069737885818257916942, 14.76447383920685856541119059477, 15.50588801950169175719704181993, 16.34549909910102467670042217157
