| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.939 + 0.342i)6-s + (−0.266 − 0.460i)7-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (−1.85 + 3.21i)11-s + (−0.499 − 0.866i)12-s + (−3.09 − 1.12i)13-s + (−0.0923 + 0.524i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.624 − 0.524i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.0776 + 0.440i)5-s + (0.383 + 0.139i)6-s + (−0.100 − 0.174i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (0.242 − 0.203i)10-s + (−0.560 + 0.970i)11-s + (−0.144 − 0.250i)12-s + (−0.857 − 0.312i)13-s + (−0.0246 + 0.140i)14-s + (−0.0448 − 0.254i)15-s + (−0.234 + 0.0855i)16-s + (−0.151 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000268658 + 0.00688996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000268658 + 0.00688996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (-4.21 + 1.10i)T \) |
| good | 7 | \( 1 + (0.266 + 0.460i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.09 + 1.12i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.624 + 0.524i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.01 + 5.74i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.68 - 6.44i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.70 + 8.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.04T + 37T^{2} \) |
| 41 | \( 1 + (9.69 - 3.52i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.768 + 4.35i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (8.35 - 7.00i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.180 + 1.02i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.09 - 4.27i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.48 - 14.1i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.91 + 3.28i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.595 + 3.37i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.02 - 0.371i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (11.8 - 4.31i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (9.02 + 15.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.60 + 1.31i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.01 - 1.69i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.06905467709118763561140947525, −10.16542238661908827279510921629, −9.806395698795572577879211609853, −8.669762961442084297447980914466, −7.35597403532259004842622683381, −7.08277466405487647410146535229, −5.56811787935747050009170629895, −4.59771913846079886700333637162, −3.30996022686348678100707883742, −2.04360545762475382680193357773,
0.00477455418305308418209864698, 1.74049952061666223572130267527, 3.51384634605734658405478140339, 5.18795135548182688485425313032, 5.55963360788584169019642869555, 6.81010597487596057924132728797, 7.62315438788011425160192913827, 8.470974536317397295818123406750, 9.438459236863552819272322851923, 10.15763901174118781869631087998
