L(s) = 1 | + (0.0380 + 1.73i)3-s + (−2.18 + 0.906i)5-s + (−1.93 − 1.93i)7-s + (−2.99 + 0.131i)9-s + (1.42 − 0.590i)11-s + (0.110 − 0.266i)13-s + (−1.65 − 3.75i)15-s + 6.17·17-s + (−7.34 − 3.04i)19-s + (3.27 − 3.41i)21-s + (−1.85 − 1.85i)23-s + (0.430 − 0.430i)25-s + (−0.342 − 5.18i)27-s + (2.11 − 5.10i)29-s − 3.42i·31-s + ⋯ |
L(s) = 1 | + (0.0219 + 0.999i)3-s + (−0.978 + 0.405i)5-s + (−0.730 − 0.730i)7-s + (−0.999 + 0.0439i)9-s + (0.429 − 0.177i)11-s + (0.0306 − 0.0739i)13-s + (−0.426 − 0.969i)15-s + 1.49·17-s + (−1.68 − 0.697i)19-s + (0.714 − 0.746i)21-s + (−0.386 − 0.386i)23-s + (0.0861 − 0.0861i)25-s + (−0.0658 − 0.997i)27-s + (0.392 − 0.948i)29-s − 0.614i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0914 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.350171 - 0.319480i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.350171 - 0.319480i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0380 - 1.73i)T \) |
good | 5 | \( 1 + (2.18 - 0.906i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (1.93 + 1.93i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.42 + 0.590i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.110 + 0.266i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 + (7.34 + 3.04i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.85 + 1.85i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2.11 + 5.10i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 3.42iT - 31T^{2} \) |
| 37 | \( 1 + (2.52 + 6.09i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.753 + 0.753i)T - 41iT^{2} \) |
| 43 | \( 1 + (1.57 + 3.79i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 1.54iT - 47T^{2} \) |
| 53 | \( 1 + (-5.12 - 12.3i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (3.08 + 7.45i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-4.28 - 1.77i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-0.531 + 1.28i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (8.72 - 8.72i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.73 + 2.73i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 + (-2.53 + 6.11i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.14 + 4.14i)T + 89iT^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−10.26914109169317202788756554086, −9.396010736601441161012995344107, −8.444461127052169342617275081841, −7.61130790069541421491266257978, −6.61414151724553005591058501460, −5.66693477461219943439179556497, −4.20102647389206336333366648476, −3.87278468221386671035175459423, −2.79299446699565377064839186476, −0.24382731706087133603410348062,
1.48721161258338831908941780796, 2.95221169791383280906943698924, 3.94305726107804760263153966694, 5.34327230477302281001963302972, 6.27059062415339959058000622596, 7.02470561419321613673192268482, 8.155927304853292600979226644604, 8.454604612833070743314316323861, 9.541304007469445729695425783164, 10.57607612292603108548657713676