| L(s) = 1 | + (2.63 + 3.01i)2-s + (−8.65 − 2.47i)3-s + (−2.15 + 15.8i)4-s + (−10.5 + 18.3i)5-s + (−15.3 − 32.5i)6-s + (−38.6 + 22.3i)7-s + (−53.4 + 35.2i)8-s + (68.7 + 42.7i)9-s + (−83.0 + 16.3i)10-s + (58.6 − 33.8i)11-s + (57.7 − 131. i)12-s + (14.5 − 25.2i)13-s + (−168. − 57.7i)14-s + (136. − 132. i)15-s + (−246. − 68.2i)16-s + 402.·17-s + ⋯ |
| L(s) = 1 | + (0.657 + 0.753i)2-s + (−0.961 − 0.274i)3-s + (−0.134 + 0.990i)4-s + (−0.423 + 0.732i)5-s + (−0.425 − 0.904i)6-s + (−0.788 + 0.455i)7-s + (−0.834 + 0.550i)8-s + (0.849 + 0.527i)9-s + (−0.830 + 0.163i)10-s + (0.485 − 0.280i)11-s + (0.401 − 0.915i)12-s + (0.0861 − 0.149i)13-s + (−0.861 − 0.294i)14-s + (0.607 − 0.588i)15-s + (−0.963 − 0.266i)16-s + 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.272195 + 1.03068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272195 + 1.03068i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.63 - 3.01i)T \) |
| 3 | \( 1 + (8.65 + 2.47i)T \) |
| good | 5 | \( 1 + (10.5 - 18.3i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (38.6 - 22.3i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-58.6 + 33.8i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-14.5 + 25.2i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 402.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 644. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (335. + 193. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (362. + 627. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-1.09e3 - 629. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.40e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (774. - 1.34e3i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.62e3 - 935. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-3.61e3 + 2.08e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 906.T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-3.91e3 - 2.26e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.31e3 + 2.27e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-58.7 - 33.8i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 1.31e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.47e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.78e3 + 2.18e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-659. + 381. i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.08e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (3.33e3 + 5.77e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−16.24153819641067455931257447480, −15.12208978289400217479018053410, −13.90967115558090436587443316189, −12.43317765287322031484985571766, −11.81547355306651402398736006706, −10.12820512639981434341097393276, −7.968335463907891955185508264157, −6.62290951581070048888091683767, −5.66444857477559475804204647984, −3.58008410780970666230741379278,
0.70367603191526192561688938739, 3.85201789034042600245905957486, 5.17957607767965929333269275992, 6.75493024986641302164522971148, 9.335447449625738492564135136123, 10.42621134549597732170899824360, 11.77990390024424673507949140877, 12.48767359197289407264790763593, 13.66786117839259718660299494128, 15.33331473749727436814699090717
