L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.814 + 3.04i)3-s + (0.499 + 0.866i)4-s + (−2.22 + 0.208i)5-s + (0.814 − 3.04i)6-s + (0.402 + 0.696i)7-s − 0.999i·8-s + (−5.98 + 3.45i)9-s + (2.03 + 0.932i)10-s + (−0.778 − 2.90i)11-s + (−2.22 + 2.22i)12-s + (0.206 + 3.59i)13-s − 0.804i·14-s + (−2.44 − 6.60i)15-s + (−0.5 + 0.866i)16-s + (6.99 + 1.87i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.470 + 1.75i)3-s + (0.249 + 0.433i)4-s + (−0.995 + 0.0931i)5-s + (0.332 − 1.24i)6-s + (0.152 + 0.263i)7-s − 0.353i·8-s + (−1.99 + 1.15i)9-s + (0.642 + 0.294i)10-s + (−0.234 − 0.876i)11-s + (−0.642 + 0.642i)12-s + (0.0571 + 0.998i)13-s − 0.215i·14-s + (−0.631 − 1.70i)15-s + (−0.125 + 0.216i)16-s + (1.69 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.495085 + 0.609361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.495085 + 0.609361i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (2.22 - 0.208i)T \) |
| 13 | \( 1 + (-0.206 - 3.59i)T \) |
good | 3 | \( 1 + (-0.814 - 3.04i)T + (-2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.402 - 0.696i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.778 + 2.90i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-6.99 - 1.87i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.51 - 1.21i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.422 + 0.113i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.58 + 2.06i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.536 + 0.536i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.482 - 0.835i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.63 + 0.437i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.64 + 6.14i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + (5.01 - 5.01i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.0422 - 0.157i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 1.93i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.82 - 2.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.63 + 9.83i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 7.75iT - 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + (-5.91 + 1.58i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (11.6 - 6.70i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.94598447331454961430182046483, −12.05777205854869509317821257732, −11.30255756336390179115109682859, −10.43684476310967409079384164074, −9.463693976157284549147439824083, −8.581790556342156993043725983889, −7.69464623653160892542532596312, −5.48959146359871244505335199626, −3.99381184012509492380932240858, −3.16793388237779430655519233887,
1.05326644270254489554451018446, 3.05061419569195185343293834072, 5.47813214442446612496689547634, 7.10636013373435566592995294194, 7.60254914359114372814537069198, 8.232036641847441072056254095583, 9.631592738551090329584692131579, 11.18703951799765987045203816490, 12.21996778639742248156125592150, 12.82703674231732484571151054494