| L(s) = 1 | + (0.602 − 1.27i)2-s + (0.427 − 0.903i)3-s + (−1.27 − 1.54i)4-s + (−2.87 − 0.426i)5-s + (−0.898 − 1.09i)6-s + (−0.214 − 2.17i)7-s + (−2.74 + 0.699i)8-s + (−0.634 − 0.773i)9-s + (−2.27 + 3.42i)10-s + (−4.29 + 0.210i)11-s + (−1.93 + 0.491i)12-s + (3.88 + 2.88i)13-s + (−2.91 − 1.03i)14-s + (−1.61 + 2.41i)15-s + (−0.757 + 3.92i)16-s + (−0.837 − 1.25i)17-s + ⋯ |
| L(s) = 1 | + (0.426 − 0.904i)2-s + (0.246 − 0.521i)3-s + (−0.636 − 0.771i)4-s + (−1.28 − 0.190i)5-s + (−0.366 − 0.445i)6-s + (−0.0810 − 0.822i)7-s + (−0.968 + 0.247i)8-s + (−0.211 − 0.257i)9-s + (−0.720 + 1.08i)10-s + (−1.29 + 0.0635i)11-s + (−0.559 + 0.141i)12-s + (1.07 + 0.799i)13-s + (−0.778 − 0.277i)14-s + (−0.416 + 0.624i)15-s + (−0.189 + 0.981i)16-s + (−0.203 − 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.320122 + 0.404785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320122 + 0.404785i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.602 + 1.27i)T \) |
| 3 | \( 1 + (-0.427 + 0.903i)T \) |
| good | 5 | \( 1 + (2.87 + 0.426i)T + (4.78 + 1.45i)T^{2} \) |
| 7 | \( 1 + (0.214 + 2.17i)T + (-6.86 + 1.36i)T^{2} \) |
| 11 | \( 1 + (4.29 - 0.210i)T + (10.9 - 1.07i)T^{2} \) |
| 13 | \( 1 + (-3.88 - 2.88i)T + (3.77 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.837 + 1.25i)T + (-6.50 + 15.7i)T^{2} \) |
| 19 | \( 1 + (-5.41 - 1.35i)T + (16.7 + 8.95i)T^{2} \) |
| 23 | \( 1 + (2.94 - 5.51i)T + (-12.7 - 19.1i)T^{2} \) |
| 29 | \( 1 + (-2.75 + 2.49i)T + (2.84 - 28.8i)T^{2} \) |
| 31 | \( 1 + (3.06 + 7.40i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (8.93 + 5.35i)T + (17.4 + 32.6i)T^{2} \) |
| 41 | \( 1 + (-0.854 + 0.259i)T + (34.0 - 22.7i)T^{2} \) |
| 43 | \( 1 + (8.48 - 4.01i)T + (27.2 - 33.2i)T^{2} \) |
| 47 | \( 1 + (11.5 + 2.29i)T + (43.4 + 17.9i)T^{2} \) |
| 53 | \( 1 + (-1.35 - 1.22i)T + (5.19 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.657 + 0.487i)T + (17.1 - 56.4i)T^{2} \) |
| 61 | \( 1 + (11.7 - 4.21i)T + (47.1 - 38.6i)T^{2} \) |
| 67 | \( 1 + (-1.37 - 3.84i)T + (-51.7 + 42.5i)T^{2} \) |
| 71 | \( 1 + (9.15 + 7.51i)T + (13.8 + 69.6i)T^{2} \) |
| 73 | \( 1 + (-0.827 + 8.40i)T + (-71.5 - 14.2i)T^{2} \) |
| 79 | \( 1 + (0.679 + 3.41i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (-8.13 + 4.87i)T + (39.1 - 73.1i)T^{2} \) |
| 89 | \( 1 + (2.34 + 4.37i)T + (-49.4 + 74.0i)T^{2} \) |
| 97 | \( 1 + (-9.28 + 3.84i)T + (68.5 - 68.5i)T^{2} \) |
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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
−9.894059387242880331923943448542, −8.890033127562427511822776813300, −7.905218141222801183321825676211, −7.37564543514354340075903504318, −6.04839975025018641218585374082, −4.90001714909131170548761545115, −3.86455278272412288614917475346, −3.26750631153160460457977702858, −1.68700875242899517392115427379, −0.21986733988900036760182752375,
3.02204496075464681526190454320, 3.50571047247677988645205140066, 4.82217737586462214256847861341, 5.42913659053082487739430662602, 6.58228562363119922063065593685, 7.62315488060758085837975997975, 8.418307817866339335535061410378, 8.647950447190895533729624326141, 10.07628846600359575018160157932, 10.91085853914560741843839282966
