| L(s) = 1 | + (0.377 − 1.36i)2-s + (−2.72 + 1.37i)3-s + (−1.71 − 1.03i)4-s + (0.348 + 0.785i)5-s + (0.839 + 4.23i)6-s + (3.08 + 1.85i)7-s + (−2.05 + 1.94i)8-s + (3.76 − 5.07i)9-s + (1.20 − 0.177i)10-s + (−4.95 − 3.86i)11-s + (6.08 + 0.456i)12-s + (−0.314 + 0.299i)13-s + (3.69 − 3.50i)14-s + (−2.02 − 1.66i)15-s + (1.87 + 3.53i)16-s + (−1.68 − 2.05i)17-s + ⋯ |
| L(s) = 1 | + (0.267 − 0.963i)2-s + (−1.57 + 0.792i)3-s + (−0.857 − 0.515i)4-s + (0.155 + 0.351i)5-s + (0.342 + 1.72i)6-s + (1.16 + 0.699i)7-s + (−0.725 + 0.688i)8-s + (1.25 − 1.69i)9-s + (0.380 − 0.0561i)10-s + (−1.49 − 1.16i)11-s + (1.75 + 0.131i)12-s + (−0.0871 + 0.0830i)13-s + (0.986 − 0.938i)14-s + (−0.523 − 0.429i)15-s + (0.469 + 0.883i)16-s + (−0.408 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00810988 + 0.177990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00810988 + 0.177990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.377 + 1.36i)T \) |
| good | 3 | \( 1 + (2.72 - 1.37i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.348 - 0.785i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-3.08 - 1.85i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (4.95 + 3.86i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.314 - 0.299i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.68 + 2.05i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.713 - 4.11i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (6.01 + 2.84i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (6.03 + 3.42i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (10.3 - 2.05i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-0.260 - 0.164i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (4.39 + 4.84i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.12 + 3.41i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-3.04 - 1.62i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (3.51 - 1.98i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (0.114 - 0.120i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.128 + 1.73i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (8.79 + 7.58i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-8.86 - 1.31i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 7.16i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (6.43 + 1.95i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 3.07i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (2.16 - 1.02i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (5.92 + 3.95i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.81051878687679706100699781563, −10.07787525279972616420724519640, −8.963718635983809001766063016643, −7.911203091515177129509551737070, −6.09199853439908669655772448755, −5.46346255711547612747164897309, −4.90610006909513985554752125884, −3.69859447583291536161919278638, −2.13992325692940047536023467486, −0.11564902616675136473463892621,
1.71324131894274705871827941383, 4.33508164067009158776121518657, 5.17594146743502866982664987883, 5.54699967832974170141860250193, 6.95497870692018962415764148368, 7.43107810072954814723888776116, 8.127039411062624303491036656582, 9.624699988737977465215747770432, 10.73201098962184140513957940733, 11.33720733742337199112522180133
