| L(s) = 1 | + (−3.57 + 0.630i)2-s + (−0.0962 + 2.99i)3-s + (8.61 − 3.13i)4-s + (1.73 + 2.06i)5-s + (−1.54 − 10.7i)6-s + (−4.85 − 8.40i)7-s + (−16.2 + 9.37i)8-s + (−8.98 − 0.577i)9-s + (−7.50 − 6.29i)10-s + 11.5i·11-s + (8.57 + 26.1i)12-s + (−0.416 − 0.349i)13-s + (22.6 + 26.9i)14-s + (−6.36 + 5.00i)15-s + (24.0 − 20.1i)16-s + (−21.3 − 25.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 + 0.315i)2-s + (−0.0320 + 0.999i)3-s + (2.15 − 0.783i)4-s + (0.346 + 0.413i)5-s + (−0.257 − 1.79i)6-s + (−0.693 − 1.20i)7-s + (−2.02 + 1.17i)8-s + (−0.997 − 0.0641i)9-s + (−0.750 − 0.629i)10-s + 1.04i·11-s + (0.714 + 2.17i)12-s + (−0.0320 − 0.0268i)13-s + (1.61 + 1.92i)14-s + (−0.424 + 0.333i)15-s + (1.50 − 1.25i)16-s + (−1.25 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0707 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0707 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0884629 - 0.0949565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0884629 - 0.0949565i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0962 - 2.99i)T \) |
| 19 | \( 1 + (18.9 - 0.162i)T \) |
| good | 2 | \( 1 + (3.57 - 0.630i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 2.06i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (4.85 + 8.40i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 11.5iT - 121T^{2} \) |
| 13 | \( 1 + (0.416 + 0.349i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (21.3 + 25.3i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (0.660 + 1.81i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (0.811 + 2.22i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 17.8T + 961T^{2} \) |
| 37 | \( 1 - 59.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (46.0 - 8.12i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (21.7 + 7.93i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (15.9 + 43.8i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (61.8 + 10.8i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-1.40 + 3.85i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-31.9 - 26.7i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (7.25 - 41.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-56.9 + 10.0i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (108. + 39.3i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (78.4 - 65.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (15.0 - 8.66i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (6.97 + 19.1i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (4.59 + 26.0i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.60592655050573884281748498522, −10.69663559155509751155462287777, −9.995084263461192828589604259780, −9.544878534554800801526120412401, −8.399126535018596678397561912064, −7.06808052554496223906746817817, −6.45835218323295809965770635115, −4.48057311246017976583661817711, −2.54257493591738472093873823607, −0.13178068227301536926394303876,
1.66154298003137122180892274631, 2.81478497480759718245464926673, 6.00286543682611050283175302088, 6.56929860053734379514461271906, 8.175160682466740931230034423523, 8.652491965593733220956944697304, 9.419514859533909814551028151473, 10.79783638838339914857394659037, 11.54938966247872654191841342241, 12.63172636024250539871132784557
