L(s) = 1 | + (2.47 + 0.664i)2-s + (3.97 + 2.29i)4-s + (0.281 + 0.281i)5-s + (−0.201 − 2.63i)7-s + (4.70 + 4.70i)8-s + (0.511 + 0.885i)10-s + (−0.939 + 3.50i)11-s + (3.44 + 1.06i)13-s + (1.25 − 6.67i)14-s + (3.95 + 6.84i)16-s + (−2.04 + 3.54i)17-s + (0.777 − 0.208i)19-s + (0.473 + 1.76i)20-s + (−4.66 + 8.07i)22-s + (4.41 − 2.54i)23-s + ⋯ |
L(s) = 1 | + (1.75 + 0.469i)2-s + (1.98 + 1.14i)4-s + (0.125 + 0.125i)5-s + (−0.0762 − 0.997i)7-s + (1.66 + 1.66i)8-s + (0.161 + 0.280i)10-s + (−0.283 + 1.05i)11-s + (0.955 + 0.294i)13-s + (0.334 − 1.78i)14-s + (0.987 + 1.71i)16-s + (−0.496 + 0.860i)17-s + (0.178 − 0.0478i)19-s + (0.105 + 0.395i)20-s + (−0.993 + 1.72i)22-s + (0.919 − 0.531i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.03653 + 1.70394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.03653 + 1.70394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.201 + 2.63i)T \) |
| 13 | \( 1 + (-3.44 - 1.06i)T \) |
good | 2 | \( 1 + (-2.47 - 0.664i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.281 - 0.281i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.939 - 3.50i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (2.04 - 3.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.777 + 0.208i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-4.41 + 2.54i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.00 + 1.74i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.44 + 4.44i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.463 + 1.73i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (0.578 - 2.15i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.99 - 5.99i)T - 47iT^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + (1.92 + 7.17i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.40 - 1.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.85 + 1.30i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.582 - 2.17i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (3.50 - 3.50i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + (5.36 + 5.36i)T + 83iT^{2} \) |
| 89 | \( 1 + (-11.5 - 3.09i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (7.71 - 2.06i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.74407454802636365838019286145, −9.658152567491778424383778185775, −8.254706182208330109071804443871, −7.36749767091159857383656913156, −6.63769181294487679871496961267, −5.99920590606828671612873893483, −4.73647848501046277919726149577, −4.21652394142574516570888781613, −3.25731884028927156124797692410, −1.93164901169315097574866402660,
1.60301737128856079626106041578, 2.99472199715788109294079453760, 3.43627367754546229081571153054, 4.91956337648341181335950137288, 5.47657949571269928486148858993, 6.18074601555878853889183398716, 7.15633018220025286438854145313, 8.535668620236582846747700220014, 9.315060503775169828015855766102, 10.62921979905567596124344813338