L(s) = 1 | + (1.86 + 1.23i)5-s + (−1.73 − 2i)7-s + (−1.5 − 2.59i)9-s + (1.5 − 2.59i)11-s − 5i·13-s + (−1.73 − i)17-s + (2.5 + 4.33i)19-s + (6.06 − 3.5i)23-s + (1.96 + 4.59i)25-s + 4·29-s + (−1 + 1.73i)31-s + (−0.767 − 5.86i)35-s + (−0.866 + 0.5i)37-s + 3·41-s − 2i·43-s + ⋯ |
L(s) = 1 | + (0.834 + 0.550i)5-s + (−0.654 − 0.755i)7-s + (−0.5 − 0.866i)9-s + (0.452 − 0.783i)11-s − 1.38i·13-s + (−0.420 − 0.242i)17-s + (0.573 + 0.993i)19-s + (1.26 − 0.729i)23-s + (0.392 + 0.919i)25-s + 0.742·29-s + (−0.179 + 0.311i)31-s + (−0.129 − 0.991i)35-s + (−0.142 + 0.0821i)37-s + 0.468·41-s − 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30087 - 0.694205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30087 - 0.694205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.86 - 1.23i)T \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 3 | \( 1 + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (6.06 - 3.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-13.8 - 8i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.58565711856057636519794869717, −9.801819370339005176273254261582, −9.037493181331962558484929333183, −7.962686256396124794997185488102, −6.72531254960470363944281516232, −6.23791852655595223827498015136, −5.21967989225295330136914365667, −3.51424921635709369385715730583, −2.95499721526651732964333005001, −0.889577415937023522694604967108,
1.76541162097098055124785547300, 2.79300864418129792760058174939, 4.52362580491294833034020751662, 5.28459725291016964542831959648, 6.34950570998274108940591651543, 7.13306027940299045540808387390, 8.555434392692556432357010655844, 9.257494658217253645225669926344, 9.689468880557694766540296011684, 10.97880363845183719545293630424