| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.480 − 1.05i)5-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−1.10 − 0.325i)10-s + (1.85 + 2.13i)11-s + (0.654 + 0.755i)12-s + (6.20 + 1.82i)13-s + (−0.415 + 0.909i)14-s + (0.972 + 0.624i)15-s + (−0.959 + 0.281i)16-s + (0.210 − 1.46i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.214 − 0.470i)5-s + (−0.0580 + 0.404i)6-s + (−0.362 + 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.350 − 0.102i)10-s + (0.558 + 0.644i)11-s + (0.189 + 0.218i)12-s + (1.71 + 0.504i)13-s + (−0.111 + 0.243i)14-s + (0.251 + 0.161i)15-s + (−0.239 + 0.0704i)16-s + (0.0510 − 0.354i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21492 - 1.05822i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21492 - 1.05822i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.38 + 4.59i)T \) |
| good | 5 | \( 1 + (0.480 + 1.05i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-1.85 - 2.13i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-6.20 - 1.82i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.210 + 1.46i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.363 + 2.52i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.556 + 3.87i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.60 + 4.24i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.38 + 3.03i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.34 + 2.94i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.78 + 5.00i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 + (-4.99 + 1.46i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-7.13 - 2.09i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.31 - 3.41i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.69 + 4.26i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (2.69 - 3.10i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.21 - 15.3i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-6.76 - 1.98i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.902 - 1.97i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.03 + 3.87i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.73 - 10.3i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.906387872402881804983015943267, −9.123522035792740357787298691228, −8.462622568696073653428990776252, −6.99530943646674308154920822548, −6.26258233366958231301726321446, −5.38097191551631645120495634770, −4.25579236573071124643528951056, −3.83023659641795131575473434111, −2.27513998261494806663278963015, −0.793569178056237950882991322215,
1.33786506600944331204063867209, 3.31976008018893423198439824133, 3.75824691554435003155019929382, 5.24540561622696750483917127215, 6.07357999822876687867769367950, 6.57255605878022507317698440954, 7.54438222595639288923030573049, 8.366219667242796067448643961653, 9.196108430398576940933251356791, 10.46584437293068830900836182688
