| L(s) = 1 | + (0.440 − 1.34i)2-s + (1.56 + 1.66i)3-s + (−1.61 − 1.18i)4-s + (−0.189 − 0.419i)5-s + (2.92 − 1.36i)6-s + (2.14 − 1.54i)7-s + (−2.30 + 1.64i)8-s + (−0.145 + 2.21i)9-s + (−0.646 + 0.0703i)10-s + (−2.63 − 4.24i)11-s + (−0.540 − 4.53i)12-s + (−0.572 − 5.80i)13-s + (−1.13 − 3.56i)14-s + (0.402 − 0.971i)15-s + (1.19 + 3.81i)16-s + (−4.42 − 0.582i)17-s + ⋯ |
| L(s) = 1 | + (0.311 − 0.950i)2-s + (0.901 + 0.962i)3-s + (−0.805 − 0.592i)4-s + (−0.0849 − 0.187i)5-s + (1.19 − 0.556i)6-s + (0.811 − 0.584i)7-s + (−0.814 + 0.580i)8-s + (−0.0484 + 0.738i)9-s + (−0.204 + 0.0222i)10-s + (−0.795 − 1.27i)11-s + (−0.156 − 1.30i)12-s + (−0.158 − 1.61i)13-s + (−0.302 − 0.953i)14-s + (0.103 − 0.250i)15-s + (0.298 + 0.954i)16-s + (−1.07 − 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31163 - 1.67246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31163 - 1.67246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.440 + 1.34i)T \) |
| 7 | \( 1 + (-2.14 + 1.54i)T \) |
| good | 3 | \( 1 + (-1.56 - 1.66i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (0.189 + 0.419i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (2.63 + 4.24i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.572 + 5.80i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (4.42 + 0.582i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (0.189 - 1.15i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-2.29 - 2.62i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-1.78 - 3.34i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-8.65 + 2.31i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.220 - 0.586i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-0.667 - 3.35i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-4.08 + 1.23i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (7.49 - 9.76i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-3.97 - 6.39i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-1.83 + 2.56i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (10.7 - 2.51i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (4.71 + 5.03i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (6.90 + 10.3i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (3.42 - 1.69i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-1.30 - 9.94i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-1.45 + 1.77i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-4.21 - 12.4i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (4.11 - 4.11i)T - 97iT^{2} \) |
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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
−10.13226047316251707401387382676, −9.100313565880592218801639472355, −8.368497504668399893216512372247, −7.897739033446133738371745731002, −6.06267283633606619938395565569, −4.95262379974047470323360843847, −4.42284360815314324679933835024, −3.22586825341099377086317612015, −2.73374438515866109110930210243, −0.852403811899941185557934839045,
1.92368358930285781693406858704, 2.74077805718215809113973137239, 4.43663435529457598174471600816, 4.92242932789178804474331092204, 6.42262146292905669410372964544, 7.07896456606315701111691683370, 7.62159378578350946124794772842, 8.693117645953238245133185673930, 8.861673471335814277442495724521, 10.10996148151261875331476823707
