L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s − 0.999·8-s + 3·10-s + (−1.5 − 2.59i)11-s + (2.5 − 4.33i)13-s + (−0.5 − 0.866i)16-s + 3·17-s − 5·19-s + (1.50 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + 5·26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s − 0.353·8-s + 0.948·10-s + (−0.452 − 0.783i)11-s + (0.693 − 1.20i)13-s + (−0.125 − 0.216i)16-s + 0.727·17-s − 1.14·19-s + (0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + 0.980·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.785916447\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.785916447\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.692023977446466306932474226306, −8.028657432039688223380371144688, −7.21296147753085255839954426964, −5.99702657260126979482580883583, −5.65264997292962608590144504757, −5.06893992920965133905414139555, −3.97090046972027519131975859418, −3.14451635807868855489452187863, −1.77934422631694402929661057909, −0.48897360983927927255896905391,
1.64738960895474321671234553994, 2.32021936395067499654888973360, 3.25271442401967611321753458479, 4.17117241775305364188435689805, 4.99825373106986070631150151506, 6.14192590752460432442474261941, 6.48474734382392155601805669638, 7.36094156930555073315845791780, 8.356314692212496075984801633281, 9.276611754769035626486610107848