| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−2.36 + 0.861i)3-s + (0.173 + 0.984i)4-s + (−0.427 + 2.42i)5-s + (2.36 + 0.861i)6-s + (1.39 + 2.41i)7-s + (0.500 − 0.866i)8-s + (2.56 − 2.15i)9-s + (1.88 − 1.58i)10-s + (0.839 − 1.45i)11-s + (−1.26 − 2.18i)12-s + (5.96 + 2.16i)13-s + (0.485 − 2.75i)14-s + (−1.07 − 6.10i)15-s + (−0.939 + 0.342i)16-s + (3.80 + 3.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−1.36 + 0.497i)3-s + (0.0868 + 0.492i)4-s + (−0.191 + 1.08i)5-s + (0.966 + 0.351i)6-s + (0.527 + 0.914i)7-s + (0.176 − 0.306i)8-s + (0.855 − 0.718i)9-s + (0.595 − 0.499i)10-s + (0.253 − 0.438i)11-s + (−0.363 − 0.630i)12-s + (1.65 + 0.601i)13-s + (0.129 − 0.735i)14-s + (−0.277 − 1.57i)15-s + (−0.234 + 0.0855i)16-s + (0.923 + 0.774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.439777 + 0.609116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439777 + 0.609116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (2.36 - 0.861i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.427 - 2.42i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.39 - 2.41i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.839 + 1.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.96 - 2.16i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.80 - 3.19i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.433 + 2.46i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.54 - 3.81i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.64 - 6.30i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.550T + 37T^{2} \) |
| 41 | \( 1 + (2.45 - 0.891i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.498 + 2.82i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.571 + 0.479i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.255 - 1.44i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.80 - 3.18i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.62 - 9.19i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.85 - 7.43i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.21 + 6.89i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.81 + 2.11i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (5.55 - 2.02i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.58 + 13.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.48 - 2.36i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (11.0 + 9.24i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.62598037383083385537301680125, −10.33630643149855150236414761178, −8.946625920991931022426194283268, −8.350579395299859162053239442621, −6.98876140068479719826374514601, −6.16714880371872929806293693335, −5.48937642800695718609043359251, −4.11637679110624444632705534918, −3.12399989965459271708223206237, −1.44997616210435665303504803927,
0.69623629257454620099399713488, 1.34195216424408510984979781064, 3.95900261107541202406887468306, 5.02118105454249139951874098670, 5.71839482983654150235220312368, 6.58936035549337308874411837138, 7.62186742209441556775810276449, 8.146374434908865567630032527978, 9.287156159258662728594975569134, 10.20855445692845922366181906934
