| L(s) = 1 | + (2.40 + 1.48i)2-s + (4.43 − 2.71i)3-s + (3.56 + 7.16i)4-s + (5.32 − 14.6i)5-s + (14.6 + 0.0808i)6-s + (−6.67 − 1.17i)7-s + (−2.08 + 22.5i)8-s + (12.3 − 24.0i)9-s + (34.5 − 27.2i)10-s + (32.8 − 11.9i)11-s + (35.2 + 22.0i)12-s + (−58.0 + 48.7i)13-s + (−14.3 − 12.7i)14-s + (−16.0 − 79.3i)15-s + (−38.5 + 51.0i)16-s + (−26.5 + 15.2i)17-s + ⋯ |
| L(s) = 1 | + (0.850 + 0.526i)2-s + (0.853 − 0.521i)3-s + (0.445 + 0.895i)4-s + (0.476 − 1.30i)5-s + (0.999 + 0.00550i)6-s + (−0.360 − 0.0635i)7-s + (−0.0920 + 0.995i)8-s + (0.455 − 0.890i)9-s + (1.09 − 0.862i)10-s + (0.900 − 0.327i)11-s + (0.847 + 0.531i)12-s + (−1.23 + 1.04i)13-s + (−0.273 − 0.243i)14-s + (−0.276 − 1.36i)15-s + (−0.602 + 0.798i)16-s + (−0.378 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.31168 - 0.0728049i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.31168 - 0.0728049i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.40 - 1.48i)T \) |
| 3 | \( 1 + (-4.43 + 2.71i)T \) |
| good | 5 | \( 1 + (-5.32 + 14.6i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (6.67 + 1.17i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-32.8 + 11.9i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (58.0 - 48.7i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (26.5 - 15.2i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-47.8 - 27.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-16.2 - 91.9i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (146. - 174. i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-134. + 23.6i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (109. + 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-42.2 - 50.2i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (37.6 + 103. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-1.93 + 10.9i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 586. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (380. + 138. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (122. - 695. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (518. + 617. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-526. - 911. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-335. + 581. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-715. + 852. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-539. - 452. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-731. - 422. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-930. + 338. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.34150392670324493156316097303, −12.48887363902764769825902594518, −11.77156307728064508796041021010, −9.451895887133853475316621615478, −8.860142976931132826222533835240, −7.52688885865759240509270583534, −6.45648640101826741944972973428, −5.01005216169471506559254736528, −3.67123627715315115155118560201, −1.81660006782353158754794988399,
2.39889653643965506782835554284, 3.23731922459860573789562955473, 4.74016031879881356592989743031, 6.34615548470218715333006739909, 7.44322090168854921783856920713, 9.469473894725188669409191274471, 10.06823068820504307919517409544, 10.99870931294477408162722674697, 12.30327135415713096725631497231, 13.49150953709013524507885059398
