| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.395 − 0.867i)3-s + (−0.959 − 0.281i)4-s + (0.0640 + 0.0739i)5-s + (0.914 − 0.268i)6-s + (0.841 + 0.540i)7-s + (0.415 − 0.909i)8-s + (1.36 − 1.58i)9-s + (−0.0822 + 0.0528i)10-s + (−0.134 − 0.933i)11-s + (0.135 + 0.943i)12-s + (2.54 − 1.63i)13-s + (−0.654 + 0.755i)14-s + (0.0387 − 0.0848i)15-s + (0.841 + 0.540i)16-s + (−0.837 + 0.245i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.228 − 0.500i)3-s + (−0.479 − 0.140i)4-s + (0.0286 + 0.0330i)5-s + (0.373 − 0.109i)6-s + (0.317 + 0.204i)7-s + (0.146 − 0.321i)8-s + (0.456 − 0.526i)9-s + (−0.0260 + 0.0167i)10-s + (−0.0404 − 0.281i)11-s + (0.0391 + 0.272i)12-s + (0.706 − 0.454i)13-s + (−0.175 + 0.201i)14-s + (0.0100 − 0.0218i)15-s + (0.210 + 0.135i)16-s + (−0.203 + 0.0596i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21850 + 0.0277523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21850 + 0.0277523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-4.61 - 1.31i)T \) |
| good | 3 | \( 1 + (0.395 + 0.867i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.0640 - 0.0739i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.134 + 0.933i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 1.63i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.837 - 0.245i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.66 - 1.66i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-5.42 + 1.59i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.38 - 3.03i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.74 - 2.01i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.88 + 5.64i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.37 - 5.19i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 + (6.92 + 4.45i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (9.32 - 5.99i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.11 - 9.02i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.06 + 7.39i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.178 - 1.24i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.56 - 1.33i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 6.82i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.28 - 8.41i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.766 + 1.67i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (9.59 + 11.0i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.77168478618521457194837272344, −10.66518661819928835588278101326, −9.628683575598436774918394817183, −8.621036913855675442196984192954, −7.74482152142092801185559303880, −6.75197185988533427164214437994, −5.93381357522998910046030301094, −4.82487288769912849243764662058, −3.33514767286789822047730570950, −1.17464639806572261952845976560,
1.52960874363064628329056203929, 3.23963283600341941539712345814, 4.51253523205491622474382633164, 5.25761503104822407935511057649, 6.87574472510791059676673451477, 7.936310222819208139425401941752, 9.124156091970467030381763466503, 9.823422897734353714467797614557, 10.91918491688815754279871843640, 11.25604877781172796757849761858
