| L(s) = 1 | + (−1.41 − 0.0380i)2-s + (0.0948 + 1.72i)3-s + (1.99 + 0.107i)4-s + (−0.342 + 0.939i)5-s + (−0.0683 − 2.44i)6-s + (0.592 + 0.104i)7-s + (−2.81 − 0.227i)8-s + (−2.98 + 0.328i)9-s + (0.519 − 1.31i)10-s + (0.464 − 0.169i)11-s + (0.00356 + 3.46i)12-s + (−2.17 + 1.82i)13-s + (−0.833 − 0.170i)14-s + (−1.65 − 0.502i)15-s + (3.97 + 0.429i)16-s + (−6.01 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0268i)2-s + (0.0547 + 0.998i)3-s + (0.998 + 0.0537i)4-s + (−0.152 + 0.420i)5-s + (−0.0279 − 0.999i)6-s + (0.223 + 0.0394i)7-s + (−0.996 − 0.0805i)8-s + (−0.994 + 0.109i)9-s + (0.164 − 0.415i)10-s + (0.140 − 0.0510i)11-s + (0.00103 + 0.999i)12-s + (−0.604 + 0.506i)13-s + (−0.222 − 0.0454i)14-s + (−0.427 − 0.129i)15-s + (0.994 + 0.107i)16-s + (−1.45 + 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0316817 + 0.518337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0316817 + 0.518337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0380i)T \) |
| 3 | \( 1 + (-0.0948 - 1.72i)T \) |
| 5 | \( 1 + (0.342 - 0.939i)T \) |
| good | 7 | \( 1 + (-0.592 - 0.104i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.464 + 0.169i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.17 - 1.82i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.01 - 3.47i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 1.47i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0877 + 0.497i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.14 - 2.56i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.719 + 0.126i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.89 + 3.44i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.41 - 6.63i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.165 + 0.937i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 + (6.43 + 2.34i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.44 + 8.22i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.49 + 4.16i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.684 + 1.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.69 - 9.86i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.20 - 10.9i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.26 - 1.05i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 7.28i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.76 + 0.642i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.02738517966019974339955993323, −10.32996171343469981395463871477, −9.450845183686899446080278482487, −8.802974289213215721428994358927, −7.894190473391518310318795960994, −6.84072988211454987437234967109, −5.87600284175658424826381507746, −4.53738261458365065072838497376, −3.35183345143746885031626350967, −2.09628896038806990256133685269,
0.38722677982382184849036843333, 1.86639737255142799879433466134, 2.99836494609053078475062748449, 4.92125487316871882391793591912, 6.10191329742227393397139704956, 7.10044399860209186810769414734, 7.66085555717151943730457817785, 8.633878234369632083450879864299, 9.241137665026908763691164330851, 10.32763970871582584772991417669
