| L(s) = 1 | + (−0.541 + 1.30i)2-s + (−0.397 − 0.265i)3-s + (−1.41 − 1.41i)4-s + (−2.19 + 0.436i)5-s + (0.562 − 0.375i)6-s + (10.6 + 2.12i)7-s + (2.61 − 1.08i)8-s + (−3.35 − 8.10i)9-s + (0.616 − 3.10i)10-s + (11.4 + 17.1i)11-s + (0.186 + 0.937i)12-s + (−11.4 + 11.4i)13-s + (−8.55 + 12.8i)14-s + (0.987 + 0.409i)15-s + 4i·16-s + (14.6 + 8.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.270 + 0.653i)2-s + (−0.132 − 0.0885i)3-s + (−0.353 − 0.353i)4-s + (−0.438 + 0.0872i)5-s + (0.0936 − 0.0626i)6-s + (1.52 + 0.303i)7-s + (0.326 − 0.135i)8-s + (−0.372 − 0.900i)9-s + (0.0616 − 0.310i)10-s + (1.03 + 1.55i)11-s + (0.0155 + 0.0781i)12-s + (−0.880 + 0.880i)13-s + (−0.611 + 0.914i)14-s + (0.0658 + 0.0272i)15-s + 0.250i·16-s + (0.859 + 0.511i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.961618 + 0.795993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961618 + 0.795993i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.541 - 1.30i)T \) |
| 5 | \( 1 + (2.19 - 0.436i)T \) |
| 17 | \( 1 + (-14.6 - 8.68i)T \) |
| good | 3 | \( 1 + (0.397 + 0.265i)T + (3.44 + 8.31i)T^{2} \) |
| 7 | \( 1 + (-10.6 - 2.12i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-11.4 - 17.1i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (11.4 - 11.4i)T - 169iT^{2} \) |
| 19 | \( 1 + (6.03 - 14.5i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-24.9 + 16.6i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-3.66 - 18.4i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (8.49 - 12.7i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (7.87 + 5.26i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-52.8 - 10.5i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (29.2 + 70.5i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-16.6 + 16.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.7 + 35.5i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (79.8 - 33.0i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-7.00 + 35.2i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 74.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (12.7 + 8.52i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (51.1 - 10.1i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-6.94 - 10.3i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-143. - 59.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (62.0 + 62.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (12.4 + 62.5i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−12.19957313569225316761953236432, −12.14521939913525375341939200197, −10.76951842388502890587945341817, −9.463460322876780395199578990852, −8.627210943629208019482344860132, −7.47067828476159495039477238365, −6.66316294020440322951568069413, −5.14821283622592559813077654579, −4.13986664525680406065619697164, −1.64229909392102398483736259796,
0.992842841963823513214449181768, 2.93044441852542084497582530857, 4.50725420940966371446686271110, 5.50708221028434668544587902216, 7.56190233810005206378529625396, 8.179194436868922009980126144094, 9.249424115676733626518431120055, 10.74029585042397675616264242101, 11.25371362642470787416208684055, 11.88898982169175481702016529182
