| L(s) = 1 | − 2-s + 4-s − i·5-s − 2.65i·7-s − 8-s + i·10-s + (−2.74 + 1.86i)11-s + 7.17i·13-s + 2.65i·14-s + 16-s − 4.82·17-s + 0.755i·19-s − i·20-s + (2.74 − 1.86i)22-s + 7.83i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447i·5-s − 1.00i·7-s − 0.353·8-s + 0.316i·10-s + (−0.827 + 0.561i)11-s + 1.98i·13-s + 0.710i·14-s + 0.250·16-s − 1.17·17-s + 0.173i·19-s − 0.223i·20-s + (0.584 − 0.397i)22-s + 1.63i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.650638 + 0.450953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.650638 + 0.450953i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (2.74 - 1.86i)T \) |
| good | 7 | \( 1 + 2.65iT - 7T^{2} \) |
| 13 | \( 1 - 7.17iT - 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 0.755iT - 19T^{2} \) |
| 23 | \( 1 - 7.83iT - 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 - 5.75T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 2.07iT - 43T^{2} \) |
| 47 | \( 1 - 8.07iT - 47T^{2} \) |
| 53 | \( 1 + 3.55iT - 53T^{2} \) |
| 59 | \( 1 + 8.41iT - 59T^{2} \) |
| 61 | \( 1 - 9.72iT - 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 2.17iT - 71T^{2} \) |
| 73 | \( 1 - 7.75iT - 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−9.895691957725307157357648004570, −9.452841362094192413345825349154, −8.547190602698300232879208768057, −7.60485330753149111793101889216, −7.00688887495891682059687304375, −6.11064432371435588215087756508, −4.68564553100735863492217598611, −4.09433088161964445099026593853, −2.45356787564394560523801387518, −1.30139547143764523110528677050,
0.48276501800401486827803001585, 2.58654373685677541256496386481, 2.85275326564997821076252045390, 4.65268888974632945287253736229, 5.79056069276133141759178151024, 6.33085986921118478027619181227, 7.54573261305548565901528587860, 8.329882235321117927176568355317, 8.768186495116050188744284059696, 9.962692383734088161617816470989
