L(s) = 1 | + (1 − 1.73i)4-s + (2.23 + 0.125i)5-s + (−1.32 − 2.29i)7-s + (2.44 + 1.41i)11-s − 2.64·13-s + (−1.99 − 3.46i)16-s + (−3.24 − 1.87i)17-s + (1.5 − 0.866i)19-s + (2.44 − 3.74i)20-s + (3.24 + 5.61i)23-s + (4.96 + 0.559i)25-s − 5.29·28-s + 1.41i·29-s + (4.5 + 2.59i)31-s + (−2.66 − 5.28i)35-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (0.998 + 0.0560i)5-s + (−0.499 − 0.866i)7-s + (0.738 + 0.426i)11-s − 0.733·13-s + (−0.499 − 0.866i)16-s + (−0.785 − 0.453i)17-s + (0.344 − 0.198i)19-s + (0.547 − 0.836i)20-s + (0.675 + 1.17i)23-s + (0.993 + 0.111i)25-s − 0.999·28-s + 0.262i·29-s + (0.808 + 0.466i)31-s + (−0.450 − 0.892i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44513 - 0.684412i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44513 - 0.684412i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.125i)T \) |
| 7 | \( 1 + (1.32 + 2.29i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + (3.24 + 1.87i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.24 - 5.61i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.96 + 2.29i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4.58iT - 43T^{2} \) |
| 47 | \( 1 + (-3.24 + 1.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.48 - 11.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.67 - 6.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9 - 5.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.9 - 6.87i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 + (8.57 + 14.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−11.33791767776135531293815928869, −10.48298348305258024863802615728, −9.691815567344530714271795565794, −9.177073410887683373936956214100, −7.21403928222603087325114013691, −6.76621534611166646491783630894, −5.65936163934638733601310939195, −4.57227526069810974224825099700, −2.78331795164472451717090859991, −1.34153190557127302939483608156,
2.15153089058037926028135939291, 3.14272419196782943660328963400, 4.74281453196294598390763398854, 6.22089068226766660334708210216, 6.64063451368173778073838193261, 8.131446045810859166554732260031, 8.990428487914217832847003537509, 9.779393797325316150312840801973, 10.97778786812831858553676178072, 11.90365430186229463705119369449