L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (1.96 + 0.714i)5-s + (−0.696 + 3.95i)7-s + (0.500 − 0.866i)8-s + (−1.04 − 1.80i)10-s + (0.199 − 0.0726i)11-s + (3.98 − 3.34i)13-s + (3.07 − 2.57i)14-s + (−0.939 + 0.342i)16-s + (1.89 + 3.27i)17-s + (0.636 − 1.10i)19-s + (−0.362 + 2.05i)20-s + (−0.199 − 0.0726i)22-s + (−0.144 − 0.816i)23-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.877 + 0.319i)5-s + (−0.263 + 1.49i)7-s + (0.176 − 0.306i)8-s + (−0.330 − 0.571i)10-s + (0.0601 − 0.0219i)11-s + (1.10 − 0.926i)13-s + (0.821 − 0.689i)14-s + (−0.234 + 0.0855i)16-s + (0.459 + 0.795i)17-s + (0.146 − 0.252i)19-s + (−0.0810 + 0.459i)20-s + (−0.0425 − 0.0154i)22-s + (−0.0300 − 0.170i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.991526 + 0.0887991i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.991526 + 0.0887991i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.96 - 0.714i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.696 - 3.95i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.199 + 0.0726i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.98 + 3.34i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.636 + 1.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.144 + 0.816i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.05 + 5.92i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.614 - 3.48i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.77 + 3.07i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.09 - 1.76i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.48 - 2.35i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.221 + 1.25i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + (8.04 + 2.92i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.492 + 2.79i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.47 + 6.27i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.86 + 10.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.375 - 0.649i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.81 + 1.52i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.73 - 7.32i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.69 + 4.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 4.29i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.79729605875323130299600197591, −11.85751387571631655462196766872, −10.76764188383350232531469575485, −9.852167193810737752539959113651, −8.939045260025894923860278058013, −8.046500745568108665553809779317, −6.30609659532932238380644180315, −5.56646102629885148838068251510, −3.34671227122929663997204631129, −2.02873927601914779256361266137,
1.40546970735566747806752543355, 3.84762208731890856364054872841, 5.41438245013610879922457912194, 6.62991842510868440245538611549, 7.49208852324356082102916366031, 8.880382319916555578585072202866, 9.727488713545481864368058432813, 10.56651170494777889007414713447, 11.63082452338115259428441745595, 13.32941605669454957754362338646