| L(s) = 1 | + (−1.28 − 0.587i)2-s + (−0.00972 + 0.0331i)3-s + (1.30 + 1.51i)4-s + (−4.99 − 0.203i)5-s + (0.0319 − 0.0368i)6-s + (0.869 − 6.04i)7-s + (−0.796 − 2.71i)8-s + (7.57 + 4.86i)9-s + (6.30 + 3.19i)10-s + (−0.789 + 0.360i)11-s + (−0.0627 + 0.0286i)12-s + (−12.0 + 1.73i)13-s + (−4.67 + 7.26i)14-s + (0.0553 − 0.163i)15-s + (−0.569 + 3.95i)16-s + (−19.6 + 22.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.293i)2-s + (−0.00324 + 0.0110i)3-s + (0.327 + 0.377i)4-s + (−0.999 − 0.0407i)5-s + (0.00532 − 0.00614i)6-s + (0.124 − 0.864i)7-s + (−0.0996 − 0.339i)8-s + (0.841 + 0.540i)9-s + (0.630 + 0.319i)10-s + (−0.0717 + 0.0327i)11-s + (−0.00523 + 0.00238i)12-s + (−0.928 + 0.133i)13-s + (−0.333 + 0.519i)14-s + (0.00368 − 0.0108i)15-s + (−0.0355 + 0.247i)16-s + (−1.15 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0830203 + 0.195786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0830203 + 0.195786i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.587i)T \) |
| 5 | \( 1 + (4.99 + 0.203i)T \) |
| 23 | \( 1 + (20.5 + 10.2i)T \) |
| good | 3 | \( 1 + (0.00972 - 0.0331i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.869 + 6.04i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (0.789 - 0.360i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (12.0 - 1.73i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (19.6 - 22.6i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (7.05 - 6.11i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (13.1 - 15.2i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (43.4 - 12.7i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (11.9 + 7.67i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (45.5 - 29.2i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-61.6 - 18.1i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 58.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (0.467 - 3.24i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (15.7 + 109. i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (26.5 + 90.4i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-16.7 + 36.7i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (31.2 - 68.3i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-66.0 + 57.2i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-66.1 + 9.51i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (25.0 + 16.0i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-4.94 + 16.8i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (112. - 72.6i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−12.42321862884073781540912513285, −10.93784322457531664880191247186, −10.70238469436069303880823894001, −9.505837114345077930405649963616, −8.241634816760248454509466401051, −7.56044385309157354271777263552, −6.67526105502262611514466976398, −4.62789313779637915514838473482, −3.78808666092419411702879018067, −1.84317595004685009630171792659,
0.13580631778651955196259137186, 2.36059882710320329381733666824, 4.12371909113799245349041889859, 5.41513335986026493678285042165, 6.93345584368132864337612904687, 7.50365009458132835786978743271, 8.767186634443258213978295551765, 9.431887814406942560283979543996, 10.61164395671256108402834711557, 11.73814458233647051903327927123
