| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.506 + 2.87i)3-s + (0.766 − 0.642i)4-s + (−1.93 − 1.62i)5-s + (0.506 + 2.87i)6-s + (2.38 + 4.13i)7-s + (0.500 − 0.866i)8-s + (−5.18 − 1.88i)9-s + (−2.37 − 0.863i)10-s + (−0.5 + 0.866i)11-s + (1.45 + 2.52i)12-s + (0.496 + 2.81i)13-s + (3.66 + 3.07i)14-s + (5.64 − 4.73i)15-s + (0.173 − 0.984i)16-s + (−4.58 + 1.66i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.292 + 1.65i)3-s + (0.383 − 0.321i)4-s + (−0.864 − 0.725i)5-s + (0.206 + 1.17i)6-s + (0.903 + 1.56i)7-s + (0.176 − 0.306i)8-s + (−1.72 − 0.628i)9-s + (−0.749 − 0.272i)10-s + (−0.150 + 0.261i)11-s + (0.421 + 0.729i)12-s + (0.137 + 0.780i)13-s + (0.978 + 0.820i)14-s + (1.45 − 1.22i)15-s + (0.0434 − 0.246i)16-s + (−1.11 + 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.866389 + 1.26234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.866389 + 1.26234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (2.25 - 3.73i)T \) |
| good | 3 | \( 1 + (0.506 - 2.87i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.93 + 1.62i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.38 - 4.13i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.496 - 2.81i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.58 - 1.66i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.69 + 3.94i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.78 - 1.37i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.76 + 3.05i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 + (-0.0850 + 0.482i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-9.44 - 7.92i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.10 + 2.22i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-7.53 + 6.32i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.47 + 2.35i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.96 + 5.84i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.30 - 1.20i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.23 - 6.90i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.10 + 11.9i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.979 - 5.55i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.20 - 5.54i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.29 + 13.0i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.854 - 0.311i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.34759389610846236406360691995, −10.97549161378269536774536223582, −9.695363206440893172967080418743, −8.787126380607764651492824915941, −8.279497592388897043689808810090, −6.32848803596997643585041212608, −5.23585420714656976361336930846, −4.60911844120874551362708451480, −3.97279235678989811349043290111, −2.36566262719764699418689559470,
0.853806723081580309605875733730, 2.60110040441814159056231112282, 3.95780032230686532854376860022, 5.18174362162221148540572569411, 6.58062489861980412408533336876, 7.21608927369791565187837775160, 7.60519649792152887948787231043, 8.523228244624748037434365858929, 10.73884163536234292909026608256, 11.11168683106739210915462459048
