L(s) = 1 | + (−0.866 + 0.5i)2-s + (2.59 + 1.5i)3-s + (0.499 − 0.866i)4-s + (−2.23 + 0.133i)5-s − 3·6-s + (−0.866 − 2.5i)7-s + 0.999i·8-s + (3 + 5.19i)9-s + (1.86 − 1.23i)10-s + (2.59 − 1.50i)12-s − 2i·13-s + (2 + 1.73i)14-s + (−6 − 3i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (−5.19 − 3i)18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (1.49 + 0.866i)3-s + (0.249 − 0.433i)4-s + (−0.998 + 0.0599i)5-s − 1.22·6-s + (−0.327 − 0.944i)7-s + 0.353i·8-s + (1 + 1.73i)9-s + (0.590 − 0.389i)10-s + (0.749 − 0.433i)12-s − 0.554i·13-s + (0.534 + 0.462i)14-s + (−1.54 − 0.774i)15-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (−1.22 − 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.829871 + 0.377410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.829871 + 0.377410i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 - 0.133i)T \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−14.99126075687247622639054563050, −14.19277905824408766246309026195, −12.98623277872393056267018527727, −11.04772205434518895150237933479, −10.13514801880762143898798322353, −9.011912729514471905709771583413, −8.047881414401046045353644578488, −7.12503350874377073437162780492, −4.48479960410171582894100632999, −3.19474381889672672190380026752,
2.27287846813021968208195845828, 3.72196179733765636182665178792, 6.71260132864708786293101035561, 7.939813405285696141072269179497, 8.630824227271925815828165466574, 9.592427995643476242217597018235, 11.47367853440099023350187217876, 12.43094751067837540466978761204, 13.27839179520853497075201283808, 14.71885130992687241874749823264