L(s) = 1 | + (−1.26 − 0.638i)2-s + (1.18 + 1.61i)4-s + (−0.380 − 0.219i)5-s + (−2.02 − 1.70i)7-s + (−0.464 − 2.79i)8-s + (0.339 + 0.519i)10-s + (1.83 − 1.05i)11-s − 3.84i·13-s + (1.46 + 3.44i)14-s + (−1.19 + 3.81i)16-s + (4.89 − 2.82i)17-s + (1.48 − 2.57i)19-s + (−0.0963 − 0.872i)20-s + (−2.98 + 0.164i)22-s + (−4.13 − 2.38i)23-s + ⋯ |
L(s) = 1 | + (−0.892 − 0.451i)2-s + (0.592 + 0.805i)4-s + (−0.170 − 0.0981i)5-s + (−0.764 − 0.644i)7-s + (−0.164 − 0.986i)8-s + (0.107 + 0.164i)10-s + (0.552 − 0.318i)11-s − 1.06i·13-s + (0.391 + 0.920i)14-s + (−0.299 + 0.954i)16-s + (1.18 − 0.684i)17-s + (0.341 − 0.591i)19-s + (−0.0215 − 0.195i)20-s + (−0.637 + 0.0350i)22-s + (−0.861 − 0.497i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.436701 - 0.530436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.436701 - 0.530436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.26 + 0.638i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.02 + 1.70i)T \) |
good | 5 | \( 1 + (0.380 + 0.219i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 1.05i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.84iT - 13T^{2} \) |
| 17 | \( 1 + (-4.89 + 2.82i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.48 + 2.57i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.13 + 2.38i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + (-3.71 - 6.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.64 + 4.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.81iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 + (0.844 - 1.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.35 - 9.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.05 - 7.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.35 - 3.09i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.79 + 3.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.16iT - 71T^{2} \) |
| 73 | \( 1 + (8.69 - 5.01i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (13.4 + 7.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 + (-9.02 - 5.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.22iT - 97T^{2} \) |
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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
−11.73922924470203646074659188052, −10.58132328070621941517738352311, −9.956746911990255376316902203085, −9.020193841528322343928567583657, −7.892121244174600465160590120775, −7.10883827049542725225442769563, −5.86175815093532902502745147218, −3.95327762730302146425938836115, −2.89180637606477860682756153213, −0.74415156774313237273644663285,
1.86023225272346241158289434204, 3.69760100500919173162550387944, 5.57463307642275145729467616432, 6.38452710401031064066498738431, 7.46272882304874651750283007291, 8.430081590058821864785374378605, 9.691341907612040194855654919646, 9.806729218936093741532895062166, 11.47729167528412266792897540695, 11.91927357813497068405119834274