L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−2 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)10-s + (−2.5 + 4.33i)11-s − 5·13-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (3.5 + 6.06i)19-s − 0.999·20-s − 5·22-s + (0.5 + 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (−0.753 + 1.30i)11-s − 1.38·13-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.802 + 1.39i)19-s − 0.223·20-s − 1.06·22-s + (0.104 + 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.121192 + 0.951022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.121192 + 0.951022i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (5.5 + 9.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.78127876532032434845701648063, −9.942536779318481670520373852285, −9.543021933622657216761220812249, −7.987044870672589076941015267156, −7.36647492212686621800083458975, −6.65236343992326758403492498746, −5.58756348665567641289927285318, −4.59177878133704356505154841170, −3.53400530488337618788129251009, −2.23024409082416265018490320927,
0.43523103010652206430414709348, 2.53113072379842688939792474628, 3.08816430600219330754148544304, 4.76693209639527399363167283380, 5.36720858753066298400408584054, 6.40845125194464362529193886438, 7.53550592955590719540367728156, 8.824237940396154554875506103122, 9.348084258811557264347194738507, 10.18428091831982305473912672176