| L(s) = 1 | + (0.908 + 3.39i)2-s + (−5.19 − 0.102i)3-s + (−3.73 + 2.15i)4-s + (−12.1 + 12.1i)5-s + (−4.37 − 17.7i)6-s + (−3.81 − 1.02i)7-s + (9.13 + 9.13i)8-s + (26.9 + 1.06i)9-s + (−52.1 − 30.1i)10-s + (−7.61 + 2.03i)11-s + (19.6 − 10.8i)12-s + (44.8 + 13.7i)13-s − 13.8i·14-s + (64.3 − 61.8i)15-s + (−39.9 + 69.1i)16-s + (17.2 + 29.8i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 1.19i)2-s + (−0.999 − 0.0197i)3-s + (−0.467 + 0.269i)4-s + (−1.08 + 1.08i)5-s + (−0.297 − 1.20i)6-s + (−0.206 − 0.0552i)7-s + (0.403 + 0.403i)8-s + (0.999 + 0.0394i)9-s + (−1.65 − 0.952i)10-s + (−0.208 + 0.0559i)11-s + (0.472 − 0.260i)12-s + (0.955 + 0.293i)13-s − 0.264i·14-s + (1.10 − 1.06i)15-s + (−0.624 + 1.08i)16-s + (0.246 + 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.132507 + 0.901673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132507 + 0.901673i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.19 + 0.102i)T \) |
| 13 | \( 1 + (-44.8 - 13.7i)T \) |
| good | 2 | \( 1 + (-0.908 - 3.39i)T + (-6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (12.1 - 12.1i)T - 125iT^{2} \) |
| 7 | \( 1 + (3.81 + 1.02i)T + (297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (7.61 - 2.03i)T + (1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (-17.2 - 29.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (26.2 - 97.9i)T + (-5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-98.5 + 170. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (60.6 + 35.0i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (70.3 + 70.3i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-38.6 - 144. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-82.0 - 306. i)T + (-5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-410. + 237. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-150. - 150. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 401. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-146. + 545. i)T + (-1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-272. - 471. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-490. + 131. i)T + (2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-138. - 37.0i)T + (3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-74.5 + 74.5i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 341.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (465. - 465. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-514. + 137. i)T + (6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-437. + 1.63e3i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−16.16058978922451449160990083129, −15.28934251060521083141899158454, −14.40382155181156002690452046020, −12.77516696596552509051710968282, −11.30440490416436550674429673706, −10.56075791990084407554378160189, −8.039991699240363717647991531091, −6.90441315587866468720661718054, −6.00303393640026246188870779636, −4.15245485083390480406075144931,
0.839980851047201055077264860826, 3.80780640918871772895438365486, 5.15816202712616528109061572337, 7.35593168145503147682985945831, 9.232384710692361695803819787961, 10.92814120109797582402909194762, 11.52359874149810540389715508068, 12.62211731545707141994024851185, 13.21750223535960825789298533254, 15.71424395128744041718416838845
