| L(s) = 1 | + (1.35 − 0.400i)2-s + (−0.871 − 1.90i)3-s + (1.67 − 1.08i)4-s + (1.73 + 2.00i)5-s + (−1.94 − 2.24i)6-s + (−0.195 − 0.125i)7-s + (1.84 − 2.14i)8-s + (−0.918 + 1.05i)9-s + (3.15 + 2.01i)10-s + (−5.40 + 0.777i)11-s + (−3.53 − 2.25i)12-s + (−0.562 − 0.876i)13-s + (−0.314 − 0.0920i)14-s + (2.30 − 5.05i)15-s + (1.64 − 3.64i)16-s + (1.07 + 3.66i)17-s + ⋯ |
| L(s) = 1 | + (0.959 − 0.282i)2-s + (−0.503 − 1.10i)3-s + (0.839 − 0.542i)4-s + (0.775 + 0.894i)5-s + (−0.794 − 0.914i)6-s + (−0.0737 − 0.0473i)7-s + (0.652 − 0.758i)8-s + (−0.306 + 0.353i)9-s + (0.996 + 0.638i)10-s + (−1.63 + 0.234i)11-s + (−1.02 − 0.652i)12-s + (−0.156 − 0.242i)13-s + (−0.0841 − 0.0245i)14-s + (0.595 − 1.30i)15-s + (0.410 − 0.911i)16-s + (0.261 + 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55915 - 0.954171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55915 - 0.954171i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 + 0.400i)T \) |
| 23 | \( 1 + (-1.59 - 4.52i)T \) |
| good | 3 | \( 1 + (0.871 + 1.90i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 2.00i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.195 + 0.125i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (5.40 - 0.777i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.562 + 0.876i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 3.66i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.315 - 1.07i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.188 - 0.641i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-7.20 - 3.29i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (6.74 - 7.78i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.98 + 3.44i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.46 + 2.95i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 3.52iT - 47T^{2} \) |
| 53 | \( 1 + (4.45 + 2.86i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.52 + 4.19i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.83 + 8.40i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (14.5 + 2.09i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (6.17 + 0.887i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (7.57 + 2.22i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.74 + 1.76i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.93 - 4.27i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.81 - 0.829i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 10.2i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.64275274385145263380951200564, −11.73053988154488005940454380140, −10.52469862818326934736309886357, −10.11248150832157844925269369657, −7.87319419888804123340543697685, −6.91085314755646243145705192434, −6.09394339068681276588536556345, −5.16697347162694861796083933619, −3.11549685329569977915460601534, −1.86560197729379970798858416510,
2.67824854603094035942337317246, 4.49383000409115450343164004647, 5.13484164801238265883512483593, 5.91297078506857972575629132753, 7.52018609902752508865228912972, 8.856540371467147968101241275453, 10.06138783851571600356684513730, 10.80178562620037812384064660154, 11.92508923365604262111357417129, 13.01041056802339397934638024315
