L(s) = 1 | + (0.781 + 0.623i)2-s + (0.405 − 1.68i)3-s + (0.222 + 0.974i)4-s + (2.67 + 1.28i)5-s + (1.36 − 1.06i)6-s + (−1.08 + 2.41i)7-s + (−0.433 + 0.900i)8-s + (−2.67 − 1.36i)9-s + (1.28 + 2.67i)10-s + (3.64 + 2.90i)11-s + (1.73 + 0.0204i)12-s + (−2.21 − 1.76i)13-s + (−2.35 + 1.21i)14-s + (3.25 − 3.98i)15-s + (−0.900 + 0.433i)16-s + (1.10 − 4.85i)17-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (0.234 − 0.972i)3-s + (0.111 + 0.487i)4-s + (1.19 + 0.576i)5-s + (0.558 − 0.434i)6-s + (−0.408 + 0.912i)7-s + (−0.153 + 0.318i)8-s + (−0.890 − 0.455i)9-s + (0.407 + 0.846i)10-s + (1.10 + 0.877i)11-s + (0.499 + 0.00590i)12-s + (−0.614 − 0.489i)13-s + (−0.628 + 0.324i)14-s + (0.840 − 1.02i)15-s + (−0.225 + 0.108i)16-s + (0.268 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02638 + 0.372920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02638 + 0.372920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 + (-0.405 + 1.68i)T \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
good | 5 | \( 1 + (-2.67 - 1.28i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-3.64 - 2.90i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.21 + 1.76i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 4.85i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + 6.81iT - 19T^{2} \) |
| 23 | \( 1 + (2.40 - 0.549i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (5.53 + 1.26i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 - 7.19iT - 31T^{2} \) |
| 37 | \( 1 + (-1.90 + 8.34i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-7.72 - 3.72i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.96 - 1.42i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (2.98 - 3.74i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (9.35 - 2.13i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.790 + 0.380i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (5.12 + 1.17i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 + (7.21 - 1.64i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 10.4i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 0.959T + 79T^{2} \) |
| 83 | \( 1 + (-0.996 - 1.25i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.90 - 6.14i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 2.75iT - 97T^{2} \) |
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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
−12.17691253802530083482638246783, −11.20307143864085379552315418672, −9.484450693162021369229386733571, −9.222115147347364642858143766828, −7.59089611054524228183303621780, −6.77483322255178909422206207047, −6.08805371631877038598024066935, −5.05399196711755033467632533806, −3.00111033859450233835074444694, −2.14063378306720586865335162501,
1.73221062145810749162858098201, 3.54456076723695872730148591029, 4.27108529902599024566066392155, 5.66287457210606214898613054387, 6.28564542754344124003432088007, 8.123828588335269751790463422497, 9.375865623190450208620809818047, 9.821904028594321770048280756658, 10.63185630825831742600469194257, 11.65725955767667373453290472087