| L(s) = 1 | + (0.672 + 0.564i)2-s + (1.45 + 0.939i)3-s + (−0.213 − 1.21i)4-s + (0.448 + 1.45i)6-s + (−0.0883 + 0.501i)7-s + (1.41 − 2.45i)8-s + (1.23 + 2.73i)9-s + (2.28 − 0.830i)11-s + (0.826 − 1.96i)12-s + (4.85 − 4.07i)13-s + (−0.342 + 0.287i)14-s + (0.0284 − 0.0103i)16-s + (−2.86 − 4.96i)17-s + (−0.711 + 2.53i)18-s + (−1.94 + 3.36i)19-s + ⋯ |
| L(s) = 1 | + (0.475 + 0.399i)2-s + (0.840 + 0.542i)3-s + (−0.106 − 0.605i)4-s + (0.183 + 0.593i)6-s + (−0.0334 + 0.189i)7-s + (0.501 − 0.868i)8-s + (0.411 + 0.911i)9-s + (0.687 − 0.250i)11-s + (0.238 − 0.566i)12-s + (1.34 − 1.13i)13-s + (−0.0914 + 0.0767i)14-s + (0.00710 − 0.00258i)16-s + (−0.695 − 1.20i)17-s + (−0.167 + 0.597i)18-s + (−0.446 + 0.772i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.59970 + 0.415257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.59970 + 0.415257i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.45 - 0.939i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.672 - 0.564i)T + (0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (0.0883 - 0.501i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.28 + 0.830i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.85 + 4.07i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.94 - 3.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.12 - 6.37i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.324 + 0.272i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 8.37i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.19 + 2.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.938 + 0.787i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.34 + 0.855i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.143 + 0.816i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 + (6.57 + 2.39i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.558 + 3.16i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.32 - 4.47i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.59 + 2.77i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.99 + 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.06 - 1.73i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.00 + 5.03i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (9.24 - 16.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.86 + 2.86i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.56324322758958343187532078861, −9.507740200810321280725437374668, −8.958598891010619217956410346956, −7.991401316640782146592650423981, −6.93133253258095848147000550859, −5.90262145227319762701542033747, −5.09218112897177596797755864837, −4.00722134489322773187217705834, −3.15302100476192062332406563698, −1.43799387542788385762720925307,
1.64972470303690108350324233630, 2.69185509699818731158871774825, 4.07831915235876894041122790245, 4.21872732713367834410619839427, 6.27373570742840591765782507147, 6.84790541669992122035163290758, 8.027971524638299229215977056485, 8.703198948520899212947848636024, 9.259839744208550793782136437631, 10.71010424055825723634660538958
