| L(s) = 1 | + (−1.51 + 2.04i)2-s + (1.98 − 0.376i)3-s + (−1.31 − 4.27i)4-s + (1.88 − 1.20i)5-s + (−2.23 + 4.63i)6-s + (−2.24 − 1.40i)7-s + (5.94 + 2.08i)8-s + (1.01 − 0.398i)9-s + (−0.375 + 5.67i)10-s + (0.550 − 1.40i)11-s + (−4.23 − 8.00i)12-s + (4.98 + 0.561i)13-s + (6.26 − 2.47i)14-s + (3.28 − 3.10i)15-s + (−5.85 + 3.98i)16-s + (7.77 + 0.291i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 + 1.44i)2-s + (1.14 − 0.217i)3-s + (−0.659 − 2.13i)4-s + (0.842 − 0.539i)5-s + (−0.911 + 1.89i)6-s + (−0.847 − 0.530i)7-s + (2.10 + 0.735i)8-s + (0.338 − 0.132i)9-s + (−0.118 + 1.79i)10-s + (0.165 − 0.422i)11-s + (−1.22 − 2.31i)12-s + (1.38 + 0.155i)13-s + (1.67 − 0.660i)14-s + (0.848 − 0.801i)15-s + (−1.46 + 0.997i)16-s + (1.88 + 0.0705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02394 + 0.401321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02394 + 0.401321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.88 + 1.20i)T \) |
| 7 | \( 1 + (2.24 + 1.40i)T \) |
| good | 2 | \( 1 + (1.51 - 2.04i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-1.98 + 0.376i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.550 + 1.40i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.98 - 0.561i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-7.77 - 0.291i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (2.46 - 4.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.146 + 3.90i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (3.66 - 0.836i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.89 - 2.24i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.01 - 0.537i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (2.67 + 5.55i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.05 - 3.00i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (2.19 + 1.61i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (3.20 + 1.69i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.154 + 2.06i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (3.26 - 10.5i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (4.71 + 1.26i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.625 - 2.73i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.99 - 1.47i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (5.16 + 2.98i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.868 - 7.71i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-3.20 - 8.16i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (0.625 + 0.625i)T + 97iT^{2} \) |
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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
−12.63856314720185331646401413804, −10.55087969179563051697007288042, −9.827825079305414700859050780658, −8.930238690214185416758353967284, −8.421242137604463780864051722066, −7.49635485463335389623591556388, −6.29224989827781826218222726085, −5.63469054359726786048588259225, −3.56423543485583277636402935431, −1.36105161679197576508298759994,
1.79821756624361193638332751714, 3.02874466383446266563044633048, 3.54260231881439966504985892064, 5.94322856368156402873994623208, 7.53166680024772319004169738821, 8.627109946029999284531356198563, 9.381729661050736817303825415408, 9.789406847403221689263284172905, 10.76281560182987941363894216728, 11.74863874739834317241991186206
