L(s) = 1 | + (−0.463 + 0.803i)2-s + (1.07 − 1.35i)3-s + (0.569 + 0.986i)4-s + (1.53 + 0.886i)5-s + (0.590 + 1.49i)6-s + (2.26 + 1.36i)7-s − 2.91·8-s + (−0.680 − 2.92i)9-s + (−1.42 + 0.822i)10-s + (−3.22 − 0.791i)11-s + (1.95 + 0.289i)12-s + 0.151i·13-s + (−2.14 + 1.19i)14-s + (2.85 − 1.12i)15-s + (0.212 − 0.368i)16-s + (−0.598 − 1.03i)17-s + ⋯ |
L(s) = 1 | + (−0.328 + 0.568i)2-s + (0.621 − 0.783i)3-s + (0.284 + 0.493i)4-s + (0.686 + 0.396i)5-s + (0.241 + 0.610i)6-s + (0.857 + 0.514i)7-s − 1.02·8-s + (−0.226 − 0.973i)9-s + (−0.450 + 0.260i)10-s + (−0.971 − 0.238i)11-s + (0.563 + 0.0836i)12-s + 0.0421i·13-s + (−0.573 + 0.318i)14-s + (0.737 − 0.291i)15-s + (0.0531 − 0.0920i)16-s + (−0.145 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38823 + 0.510404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38823 + 0.510404i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.07 + 1.35i)T \) |
| 7 | \( 1 + (-2.26 - 1.36i)T \) |
| 11 | \( 1 + (3.22 + 0.791i)T \) |
good | 2 | \( 1 + (0.463 - 0.803i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 0.886i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 0.151iT - 13T^{2} \) |
| 17 | \( 1 + (0.598 + 1.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.13 - 2.38i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.52 - 0.880i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + (2.85 + 4.95i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.12 - 7.14i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + (-1.28 - 0.739i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.94 + 2.27i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.850 - 0.490i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.24 + 3.02i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.75 + 11.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.53iT - 71T^{2} \) |
| 73 | \( 1 + (9.61 - 5.55i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.02 - 4.63i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.653T + 83T^{2} \) |
| 89 | \( 1 + (-0.343 - 0.198i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−12.23193988181191722066636114934, −11.63005635131676018838109973277, −10.22659514176497328491787023870, −8.988869966001429476454879117817, −8.195041474415099723841045287490, −7.48106757860382923288516948868, −6.44927025432118330677278820404, −5.40089237305695167309652254376, −3.17212336832817533793396821672, −2.11807063663803369736271006844,
1.68141957730140076029746400938, 2.99541541897128904032819822924, 4.77947313114691023970130980024, 5.52750491180248734966486824517, 7.27449354837838741902518230428, 8.512139788851842355890369287142, 9.350338514525173396493396762592, 10.30464132519296240414206791132, 10.72469896715252643572178924067, 11.79447679410188751116047791289