L(s) = 1 | + (−1.28 − 0.587i)2-s + (1.62 − 5.54i)3-s + (1.30 + 1.51i)4-s + (4.18 + 2.73i)5-s + (−5.35 + 6.17i)6-s + (0.543 − 3.77i)7-s + (−0.796 − 2.71i)8-s + (−20.5 − 13.1i)9-s + (−3.77 − 5.97i)10-s + (15.1 − 6.92i)11-s + (10.5 − 4.80i)12-s + (−6.09 + 0.876i)13-s + (−2.91 + 4.54i)14-s + (21.9 − 18.7i)15-s + (−0.569 + 3.95i)16-s + (−8.15 + 9.40i)17-s + ⋯ |
L(s) = 1 | + (−0.643 − 0.293i)2-s + (0.542 − 1.84i)3-s + (0.327 + 0.377i)4-s + (0.836 + 0.547i)5-s + (−0.892 + 1.02i)6-s + (0.0775 − 0.539i)7-s + (−0.0996 − 0.339i)8-s + (−2.28 − 1.46i)9-s + (−0.377 − 0.597i)10-s + (1.37 − 0.629i)11-s + (0.876 − 0.400i)12-s + (−0.468 + 0.0674i)13-s + (−0.208 + 0.324i)14-s + (1.46 − 1.25i)15-s + (−0.0355 + 0.247i)16-s + (−0.479 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.546858 - 1.41744i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.546858 - 1.41744i\) |
\(L(2)\) |
|
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 | \( 1 + (-4.18 - 2.73i)T \) |
| 23 | \( 1 + (1.00 + 22.9i)T \) |
good | 3 | \( 1 + (-1.62 + 5.54i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-0.543 + 3.77i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-15.1 + 6.92i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (6.09 - 0.876i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (8.15 - 9.40i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-18.5 + 16.1i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (26.1 - 30.1i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (37.5 - 11.0i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-17.7 - 11.4i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-29.7 + 19.0i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-3.53 - 1.03i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 20.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-2.70 + 18.7i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (7.82 + 54.4i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-27.3 - 93.2i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (30.8 - 67.4i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-15.0 + 33.0i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-22.4 + 19.4i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-83.0 + 11.9i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 7.84i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-31.9 + 108. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-142. + 91.6i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−11.61098755689900516904976669117, −10.83397947332896059559334269600, −9.290082897494235033532483804239, −8.762315257593175239634825319810, −7.38936732955418449871821941251, −6.88866653885677708208022899391, −5.97998714663523634015325953796, −3.34292383088434896017821497287, −2.11800322631363610647551434812, −1.00831431906450097520145022460,
2.17014934888879424633238138611, 3.86125907172489073787309633196, 5.08529650544619708966618913529, 5.90780043716744664678090904304, 7.71319403004049262052266502691, 9.006314353686937364302941427297, 9.461574142591931063563418329590, 9.810820038846857804854546798864, 11.10512389575186263271669477254, 12.00550726768855843146832269273