| L(s) = 1 | + (−0.641 − 1.67i)2-s + (0.0377 − 0.719i)3-s + (−0.891 + 0.802i)4-s + (−0.618 − 2.14i)5-s + (−1.22 + 0.398i)6-s + (−2.17 + 1.50i)7-s + (−1.27 − 0.649i)8-s + (2.46 + 0.259i)9-s + (−3.19 + 2.41i)10-s + (−0.433 − 4.12i)11-s + (0.544 + 0.672i)12-s + (0.180 + 1.14i)13-s + (3.90 + 2.67i)14-s + (−1.57 + 0.364i)15-s + (−0.518 + 4.93i)16-s + (−4.26 + 2.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.453 − 1.18i)2-s + (0.0217 − 0.415i)3-s + (−0.445 + 0.401i)4-s + (−0.276 − 0.960i)5-s + (−0.500 + 0.162i)6-s + (−0.822 + 0.568i)7-s + (−0.450 − 0.229i)8-s + (0.822 + 0.0864i)9-s + (−1.00 + 0.762i)10-s + (−0.130 − 1.24i)11-s + (0.157 + 0.194i)12-s + (0.0501 + 0.316i)13-s + (1.04 + 0.713i)14-s + (−0.405 + 0.0940i)15-s + (−0.129 + 1.23i)16-s + (−1.03 + 0.671i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0222094 - 0.783039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0222094 - 0.783039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.618 + 2.14i)T \) |
| 7 | \( 1 + (2.17 - 1.50i)T \) |
| good | 2 | \( 1 + (0.641 + 1.67i)T + (-1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (-0.0377 + 0.719i)T + (-2.98 - 0.313i)T^{2} \) |
| 11 | \( 1 + (0.433 + 4.12i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.180 - 1.14i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (4.26 - 2.76i)T + (6.91 - 15.5i)T^{2} \) |
| 19 | \( 1 + (-3.44 + 3.82i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-6.12 + 2.34i)T + (17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-0.495 - 0.161i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.169 + 0.795i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.99 + 1.61i)T + (7.69 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-1.80 + 2.49i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.25 + 8.25i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.95 + 3.01i)T + (-19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (-12.0 - 0.631i)T + (52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (4.30 - 1.91i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (5.80 - 13.0i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (3.05 + 4.69i)T + (-27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (3.20 - 9.85i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (9.44 - 11.6i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.867 - 4.08i)T + (-72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (1.23 - 2.42i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (4.81 + 2.14i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (1.71 + 3.35i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.18116928262919902630580779758, −11.32290123509660792850942487738, −10.34161235321996911472895211427, −9.018134714122926281976182739240, −8.804827081674033415626080902987, −7.04978691106476855682816829853, −5.77379978188016697184095709078, −4.05183460245532029867171524377, −2.57534201478560802995894278734, −0.866463512295483546970962517640,
3.08494505313068533370686271977, 4.58063363947400830058682022351, 6.22898633301203360836921794690, 7.22747613769569363706814766156, 7.51297882374839424133402353024, 9.295261729669829134394524941724, 9.907273424478746140454866913611, 10.95364309430514379371816115829, 12.26676243591138274124553629821, 13.38626145435136864594841381815
