| L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.11 − 1.32i)3-s + (−0.959 + 0.281i)4-s + (1.06 − 1.96i)5-s + (−1.47 − 0.909i)6-s + (0.483 − 0.310i)7-s + (0.415 + 0.909i)8-s + (−0.535 − 2.95i)9-s + (−2.09 − 0.776i)10-s + (0.617 − 4.29i)11-s + (−0.690 + 1.58i)12-s + (−1.02 + 1.59i)13-s + (−0.376 − 0.434i)14-s + (−1.42 − 3.60i)15-s + (0.841 − 0.540i)16-s + (−1.19 + 4.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.640 − 0.767i)3-s + (−0.479 + 0.140i)4-s + (0.477 − 0.878i)5-s + (−0.601 − 0.371i)6-s + (0.182 − 0.117i)7-s + (0.146 + 0.321i)8-s + (−0.178 − 0.983i)9-s + (−0.663 − 0.245i)10-s + (0.186 − 1.29i)11-s + (−0.199 + 0.458i)12-s + (−0.284 + 0.442i)13-s + (−0.100 − 0.116i)14-s + (−0.368 − 0.929i)15-s + (0.210 − 0.135i)16-s + (−0.289 + 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.409892 - 1.70774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.409892 - 1.70774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-1.11 + 1.32i)T \) |
| 5 | \( 1 + (-1.06 + 1.96i)T \) |
| 23 | \( 1 + (-4.22 - 2.26i)T \) |
| good | 7 | \( 1 + (-0.483 + 0.310i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.617 + 4.29i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.02 - 1.59i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.19 - 4.06i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.213 - 0.726i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.29 + 4.39i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.13 - 2.48i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.469 + 0.541i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.688 - 0.596i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.91 + 4.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + (-1.10 - 1.72i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.47 - 3.85i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.92 + 1.79i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 10.4i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-11.6 + 1.67i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (1.52 + 5.17i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.12 - 7.97i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.24 + 7.14i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.53 + 9.92i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.09 - 5.87i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.972325778534408575610098833019, −9.071865071355762607009202198731, −8.575179154092295938371799472507, −7.82282110148581254910865208413, −6.51491822077446416044362889964, −5.61144142321301116918540547237, −4.31720840639891127524449708148, −3.23189764853999355146240384385, −1.97255389455829481136035684630, −0.939454280058513694149898204746,
2.20746758250743921516479751647, 3.26621438131262781374800315457, 4.63602448792391813043490553012, 5.25019301740037030931432081094, 6.64264737981635657721921823003, 7.29024503096251405284223635742, 8.202732726709331178955334065136, 9.316323184637992454860264945241, 9.726092889353088883517742427051, 10.55081358541927037412992981053
