| L(s) = 1 | + (0.557 + 1.29i)2-s + (−1.59 − 0.667i)3-s + (−1.37 + 1.44i)4-s + (−1.27 + 3.49i)5-s + (−0.0226 − 2.44i)6-s + (1.63 + 0.287i)7-s + (−2.65 − 0.986i)8-s + (2.10 + 2.13i)9-s + (−5.24 + 0.293i)10-s + (−0.526 + 0.191i)11-s + (3.17 − 1.39i)12-s + (1.99 − 1.66i)13-s + (0.534 + 2.28i)14-s + (4.36 − 4.73i)15-s + (−0.195 − 3.99i)16-s + (4.50 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.393 + 0.919i)2-s + (−0.922 − 0.385i)3-s + (−0.689 + 0.724i)4-s + (−0.568 + 1.56i)5-s + (−0.00924 − 0.999i)6-s + (0.616 + 0.108i)7-s + (−0.937 − 0.348i)8-s + (0.702 + 0.711i)9-s + (−1.65 + 0.0927i)10-s + (−0.158 + 0.0577i)11-s + (0.915 − 0.402i)12-s + (0.551 − 0.463i)13-s + (0.142 + 0.609i)14-s + (1.12 − 1.22i)15-s + (−0.0487 − 0.998i)16-s + (1.09 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.387758 + 0.745792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.387758 + 0.745792i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.557 - 1.29i)T \) |
| 3 | \( 1 + (1.59 + 0.667i)T \) |
| good | 5 | \( 1 + (1.27 - 3.49i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.526 - 0.191i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.99 + 1.66i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.50 + 2.60i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.925 + 0.534i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.54 - 8.77i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.702 - 0.837i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 0.559i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 5.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.556 + 0.662i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.690 - 1.89i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 6.95i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.80iT - 53T^{2} \) |
| 59 | \( 1 + (8.57 + 3.11i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.832 + 4.71i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.06 - 8.42i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.98 + 6.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.92 + 3.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.28 + 7.49i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.28 - 1.08i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (10.9 + 6.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.74 + 1.36i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.17934119292723388174488371655, −13.22480071246296518009960791768, −11.84661529423290315104139196255, −11.25719498724101914094748495868, −9.992877123835781089131139259885, −7.956157687569825371641507699014, −7.32364598547676916236289044724, −6.26121208027768024726773744793, −5.12187465075092575646126382043, −3.38466821054759219484089449232,
1.08546955180161259057205667368, 4.09272989351351807100539236268, 4.80573137082192001665566467417, 5.95349886401429070944381430074, 8.204736652227884239036503543006, 9.219136988520783328913035665596, 10.49356756671952273814436585212, 11.38380042200304928248017810844, 12.37288456887227473714391443561, 12.74970611903526583852844369445
