| 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
−15.71424395128744041718416838845, −13.21750223535960825789298533254, −12.62211731545707141994024851185, −11.52359874149810540389715508068, −10.92814120109797582402909194762, −9.232384710692361695803819787961, −7.35593168145503147682985945831, −5.15816202712616528109061572337, −3.80780640918871772895438365486, −0.839980851047201055077264860826,
4.15245485083390480406075144931, 6.00303393640026246188870779636, 6.90441315587866468720661718054, 8.039991699240363717647991531091, 10.56075791990084407554378160189, 11.30440490416436550674429673706, 12.77516696596552509051710968282, 14.40382155181156002690452046020, 15.28934251060521083141899158454, 16.16058978922451449160990083129
