| L(s) = 1 | + (−0.819 − 0.573i)2-s + (1.72 + 0.116i)3-s + (0.342 + 0.939i)4-s + (2.10 + 0.759i)5-s + (−1.34 − 1.08i)6-s + (−0.894 − 1.91i)7-s + (0.258 − 0.965i)8-s + (2.97 + 0.403i)9-s + (−1.28 − 1.82i)10-s + (−2.96 − 3.53i)11-s + (0.481 + 1.66i)12-s + (3.86 + 5.51i)13-s + (−0.367 + 2.08i)14-s + (3.54 + 1.55i)15-s + (−0.766 + 0.642i)16-s + (0.169 + 0.632i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (0.997 + 0.0673i)3-s + (0.171 + 0.469i)4-s + (0.940 + 0.339i)5-s + (−0.550 − 0.443i)6-s + (−0.337 − 0.724i)7-s + (0.0915 − 0.341i)8-s + (0.990 + 0.134i)9-s + (−0.406 − 0.578i)10-s + (−0.895 − 1.06i)11-s + (0.138 + 0.480i)12-s + (1.07 + 1.52i)13-s + (−0.0981 + 0.556i)14-s + (0.915 + 0.402i)15-s + (−0.191 + 0.160i)16-s + (0.0410 + 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44693 - 0.273540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44693 - 0.273540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (-1.72 - 0.116i)T \) |
| 5 | \( 1 + (-2.10 - 0.759i)T \) |
| good | 7 | \( 1 + (0.894 + 1.91i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (2.96 + 3.53i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.86 - 5.51i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.169 - 0.632i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.874 - 0.504i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.82 + 1.78i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.766 + 4.34i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.38 + 1.23i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (5.31 - 1.42i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (10.6 + 1.88i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.247 - 2.82i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (1.70 - 0.796i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-5.85 - 5.85i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.33 - 1.12i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.87 + 2.50i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (11.3 - 7.93i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.31 - 1.33i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (13.9 + 3.73i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.93 - 0.518i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (8.78 - 12.5i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (6.23 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 0.979i)T + (95.5 + 16.8i)T^{2} \) |
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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
−11.65876050709927424564671383857, −10.50276785546760876685173025431, −10.08855735415807149045425191531, −8.968381925859226512280223926310, −8.327611218774092393146839735048, −7.07764778342444605817389971510, −6.11214096655044138750214768339, −4.14680207813734709919968366996, −3.02475766564405094259943734781, −1.71922684506479940035877011886,
1.80613965576274427431558678792, 3.05406497845878278076728254437, 5.03884747508358476397172781977, 6.02813446460704645945497323964, 7.27804066078123156074715334497, 8.348922663873026971897017387038, 8.921564895794841469662338868600, 10.06651941832333911344884292498, 10.36075701497514441135834049398, 12.27085570838162013386564458526
