L(s) = 1 | + (0.928 − 1.10i)2-s + (1.07 − 1.35i)3-s + (−0.0152 − 0.0867i)4-s + (0.0427 + 2.23i)5-s + (−0.508 − 2.45i)6-s + (1.83 − 1.06i)7-s + (2.39 + 1.38i)8-s + (−0.696 − 2.91i)9-s + (2.51 + 2.02i)10-s + (−3.86 − 2.23i)11-s + (−0.134 − 0.0722i)12-s + (0.689 + 0.250i)13-s + (0.532 − 3.01i)14-s + (3.08 + 2.34i)15-s + (3.91 − 1.42i)16-s + (−0.461 − 0.387i)17-s + ⋯ |
L(s) = 1 | + (0.656 − 0.782i)2-s + (0.619 − 0.784i)3-s + (−0.00764 − 0.0433i)4-s + (0.0191 + 0.999i)5-s + (−0.207 − 1.00i)6-s + (0.694 − 0.400i)7-s + (0.845 + 0.488i)8-s + (−0.232 − 0.972i)9-s + (0.795 + 0.641i)10-s + (−1.16 − 0.673i)11-s + (−0.0387 − 0.0208i)12-s + (0.191 + 0.0696i)13-s + (0.142 − 0.806i)14-s + (0.796 + 0.604i)15-s + (0.979 − 0.356i)16-s + (−0.111 − 0.0939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.89734 - 1.14808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89734 - 1.14808i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.07 + 1.35i)T \) |
| 5 | \( 1 + (-0.0427 - 2.23i)T \) |
| 19 | \( 1 + (3.57 - 2.48i)T \) |
good | 2 | \( 1 + (-0.928 + 1.10i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.83 + 1.06i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.86 + 2.23i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.689 - 0.250i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.461 + 0.387i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.625 - 3.54i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.26 + 1.89i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.03 - 0.596i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + (9.15 - 3.33i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.1 - 1.79i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.71 + 3.95i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.54 + 1.68i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.54 - 5.49i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.77 + 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.62 + 3.88i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.01 - 5.78i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.863 - 2.37i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.52 + 9.69i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.00 - 6.93i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.94 + 2.16i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.69 + 7.29i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.70905397032089335372786221455, −10.92765272036871641755501557107, −10.23034687619892942072570880173, −8.489017851285111134279800013133, −7.80179147801775633297037747929, −6.93061962079719216224515470219, −5.52228486741033745506701466041, −3.91845657140520846949806755639, −2.97024396576419581798622958304, −1.92324435807454003401316899236,
2.15110587153791109471198231131, 4.14394736649681782409535930120, 4.97161487663287318963750496039, 5.50253215154457495200004385180, 7.14589907479791435630575103723, 8.252344424817212763457120570822, 8.864777408688187647581514185416, 10.16335759586387091978821620612, 10.79094304642141070883363946297, 12.30403371122466550061601962638