| L(s) = 1 | + (−0.888 + 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (−1.11 + 1.06i)5-s + (0.415 − 0.909i)6-s + (−2.44 + 1.01i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.504 − 1.45i)10-s + (0.147 + 0.0762i)11-s + (0.0475 + 0.998i)12-s + (−4.11 − 4.74i)13-s + (1.70 − 2.02i)14-s + (0.219 − 1.52i)15-s + (−0.327 − 0.945i)16-s + (6.12 − 0.585i)17-s + ⋯ |
| L(s) = 1 | + (−0.628 + 0.324i)2-s + (−0.453 + 0.356i)3-s + (0.290 − 0.407i)4-s + (−0.499 + 0.476i)5-s + (0.169 − 0.371i)6-s + (−0.922 + 0.385i)7-s + (−0.0503 + 0.349i)8-s + (0.0785 − 0.323i)9-s + (0.159 − 0.461i)10-s + (0.0445 + 0.0229i)11-s + (0.0137 + 0.288i)12-s + (−1.14 − 1.31i)13-s + (0.455 − 0.541i)14-s + (0.0567 − 0.394i)15-s + (−0.0817 − 0.236i)16-s + (1.48 − 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.579198 - 0.0315771i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.579198 - 0.0315771i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.888 - 0.458i)T \) |
| 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (2.44 - 1.01i)T \) |
| 23 | \( 1 + (4.58 - 1.39i)T \) |
| good | 5 | \( 1 + (1.11 - 1.06i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (-0.147 - 0.0762i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (4.11 + 4.74i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.12 + 0.585i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (1.33 + 0.127i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (-4.31 + 9.44i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-3.60 + 1.44i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (1.47 - 6.06i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (-5.43 - 1.59i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.30 - 9.09i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-2.69 + 4.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.96 + 0.571i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-3.09 + 8.93i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (2.31 + 1.82i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (-0.403 + 8.47i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (-11.7 - 7.55i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-8.08 + 11.3i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (7.95 - 1.53i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-7.94 + 2.33i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-4.56 - 1.82i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (-2.57 - 0.755i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.861076779300927765921824920727, −9.567462297925827068533848690765, −8.022585988070656913689478639638, −7.73664854686980232522724889847, −6.53775157529078277974596380263, −5.88221799890681814585003077655, −4.94770122225789772332608535879, −3.53879696526808809770664214053, −2.63723860180954489634501049443, −0.48651581731318302348971495025,
0.881674123297842101720262285798, 2.35327864912578501917377025685, 3.69366041919986929102810003096, 4.62571496386763605212780761067, 5.89081258083654808496529667701, 6.90414423986250947920991718350, 7.42752443642920998293402050522, 8.427314365423201342087859142234, 9.275728845161405641735270868539, 10.08856567229222514707975870511
