| L(s) = 1 | + (−0.267 + 0.318i)2-s + (−1.72 + 0.159i)3-s + (0.317 + 1.79i)4-s + (0.409 − 0.591i)6-s + (1.29 + 0.229i)7-s + (−1.37 − 0.795i)8-s + (2.94 − 0.551i)9-s + (4.90 − 1.78i)11-s + (−0.834 − 3.05i)12-s + (−0.0116 − 0.0138i)13-s + (−0.419 + 0.352i)14-s + (−2.81 + 1.02i)16-s + (2.71 − 1.56i)17-s + (−0.612 + 1.08i)18-s + (0.208 − 0.361i)19-s + ⋯ |
| L(s) = 1 | + (−0.188 + 0.225i)2-s + (−0.995 + 0.0922i)3-s + (0.158 + 0.899i)4-s + (0.167 − 0.241i)6-s + (0.491 + 0.0866i)7-s + (−0.486 − 0.281i)8-s + (0.982 − 0.183i)9-s + (1.47 − 0.537i)11-s + (−0.241 − 0.881i)12-s + (−0.00321 − 0.00383i)13-s + (−0.112 + 0.0941i)14-s + (−0.703 + 0.256i)16-s + (0.658 − 0.379i)17-s + (−0.144 + 0.255i)18-s + (0.0478 − 0.0829i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.952697 + 0.656169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.952697 + 0.656169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 - 0.159i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.267 - 0.318i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.29 - 0.229i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.90 + 1.78i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0116 + 0.0138i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.71 + 1.56i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.208 + 0.361i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 0.179i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.98 - 5.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.647 - 3.67i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (3.83 - 2.21i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.84 - 7.80i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.99 - 1.23i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 1.30iT - 53T^{2} \) |
| 59 | \( 1 + (3.47 + 1.26i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.08 - 8.44i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.473 - 0.273i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.374 + 0.314i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.96 - 3.53i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.40 - 9.34i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.89883331943212527691225917247, −9.753779261489044242392957440414, −8.881327715195797971850836825456, −8.040117042328241503195326948155, −6.93028473831559194781934470117, −6.46074014022514838127879694262, −5.24995819315252321586810110805, −4.22726144419714440726639432746, −3.18315294689791968042535083167, −1.24223515197430750867408353442,
0.944414830561519048184051753636, 1.96176927641636137961216743588, 3.99763362034652723826169706550, 4.94437995554437816256947816762, 5.90664027789556742310642621173, 6.58923841618203006434254658260, 7.50389829580988638938207523338, 8.835969069202816542768988443299, 9.743679181354414770271210992640, 10.37368161777465743378861425113
