| L(s) = 1 | + (−4.36 − 2.44i)5-s + 2.79i·7-s + 18.1i·11-s − 23.0i·13-s + 5.72·17-s + 23.1·19-s + 0.271·23-s + (13.0 + 21.2i)25-s − 39.7i·29-s + 47.3·31-s + (6.82 − 12.2i)35-s − 34.8i·37-s − 13.2i·41-s − 46.7i·43-s − 40.9·47-s + ⋯ |
| L(s) = 1 | + (−0.872 − 0.488i)5-s + 0.399i·7-s + 1.64i·11-s − 1.77i·13-s + 0.336·17-s + 1.22·19-s + 0.0118·23-s + (0.523 + 0.851i)25-s − 1.37i·29-s + 1.52·31-s + (0.194 − 0.348i)35-s − 0.942i·37-s − 0.323i·41-s − 1.08i·43-s − 0.870·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.458599510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458599510\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.36 + 2.44i)T \) |
| good | 7 | \( 1 - 2.79iT - 49T^{2} \) |
| 11 | \( 1 - 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 23.0iT - 169T^{2} \) |
| 17 | \( 1 - 5.72T + 289T^{2} \) |
| 19 | \( 1 - 23.1T + 361T^{2} \) |
| 23 | \( 1 - 0.271T + 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 - 47.3T + 961T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 91.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 78.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.91iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 63.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.284T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 92.9iT - 9.40e3T^{2} \) |
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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
−10.35960695316802119939884389210, −9.783249010540094298896499855454, −8.661414007539616987532208570800, −7.76278599163881387647142556416, −7.24486505499314037530679437588, −5.70298091042920946086717120353, −4.91146452031773010622554800347, −3.82051151121896728239863977875, −2.54622887259097882223733260228, −0.76760870114311455137911986603,
1.02290069279573665269526890177, 2.98460700410702451579507870099, 3.80559388466228488940765137014, 4.93343453075699766950193652077, 6.31335588900175068848086073253, 7.01275670838954465020833305254, 8.035579620599367863590679104455, 8.787426955882121379425837123471, 9.845431488513289114133920409654, 10.87818185207270490287712907947
