| L(s) = 1 | + 2.96·3-s + i·5-s + (−0.337 − 2.62i)7-s + 5.77·9-s + 3.63i·11-s + 4.77i·13-s + 2.96i·15-s − 4.77i·17-s + 4.57·19-s + (−1 − 7.77i)21-s − 5.24i·23-s − 25-s + 8.20·27-s − 4.77·29-s + 5.92·31-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 0.447i·5-s + (−0.127 − 0.991i)7-s + 1.92·9-s + 1.09i·11-s + 1.32i·13-s + 0.764i·15-s − 1.15i·17-s + 1.04·19-s + (−0.218 − 1.69i)21-s − 1.09i·23-s − 0.200·25-s + 1.58·27-s − 0.886·29-s + 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.52204 + 0.161585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52204 + 0.161585i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.337 + 2.62i)T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 11 | \( 1 - 3.63iT - 11T^{2} \) |
| 13 | \( 1 - 4.77iT - 13T^{2} \) |
| 17 | \( 1 + 4.77iT - 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 5.24iT - 23T^{2} \) |
| 29 | \( 1 + 4.77T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 + 3.54iT - 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 7.27iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 6.85iT - 79T^{2} \) |
| 83 | \( 1 - 2.02T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 16.7iT - 97T^{2} \) |
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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
−10.46776409090982828144678196067, −9.669841680231414007355221038015, −9.196772755857424269281238340938, −8.067589268879347754786276263263, −7.12058046289015186792627431310, −6.90713481072919910615991601902, −4.75575758033386263790945343447, −3.91574118070952209484185765662, −2.89307647508808994445479466494, −1.77872985195593248326968872668,
1.63473736122001056459159440224, 3.06080708187021258242744351199, 3.51611998449194819424859069563, 5.16995450696346502179408614633, 6.09093310001214468960673610577, 7.66968035251993757244555327392, 8.202522569752467236935725158640, 8.872251905320772611874186025062, 9.561478304751551990920149835806, 10.49565231983612103459568068477
