L(s) = 1 | + (−2.05 + 1.18i)2-s + (1.81 − 3.14i)4-s + 3.43·5-s + 3.86i·8-s + (−7.05 + 4.07i)10-s + 0.313i·11-s + (5.09 − 2.94i)13-s + (−0.958 − 1.65i)16-s + (−0.476 − 0.825i)17-s + (1.09 + 0.630i)19-s + (6.23 − 10.7i)20-s + (−0.372 − 0.645i)22-s + 6.82i·23-s + 6.80·25-s + (−6.98 + 12.0i)26-s + ⋯ |
L(s) = 1 | + (−1.45 + 0.838i)2-s + (0.907 − 1.57i)4-s + 1.53·5-s + 1.36i·8-s + (−2.23 + 1.28i)10-s + 0.0946i·11-s + (1.41 − 0.816i)13-s + (−0.239 − 0.414i)16-s + (−0.115 − 0.200i)17-s + (0.250 + 0.144i)19-s + (1.39 − 2.41i)20-s + (−0.0794 − 0.137i)22-s + 1.42i·23-s + 1.36·25-s + (−1.36 + 2.37i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.164694510\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164694510\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.05 - 1.18i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 11 | \( 1 - 0.313iT - 11T^{2} \) |
| 13 | \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.476 + 0.825i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.630i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.82iT - 23T^{2} \) |
| 29 | \( 1 + (3.43 + 1.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.53 - 2.61i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.68 - 4.65i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0699 - 0.121i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.00 + 1.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 5.98i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.824 + 1.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.57 + 1.48i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-0.354 + 0.204i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.23 + 9.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.00 - 6.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.05 + 1.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.642080397915973832207090419267, −8.912580308206595624465314761015, −8.280893716584021875377613135063, −7.37637075967698641302868101531, −6.48535046932438669793161806159, −5.86759825574341153916364136103, −5.26283623122331134265705341367, −3.44395091341314361948001220262, −1.96002037266403129656796687931, −1.05520005234121178674095857737,
1.05874354140039303239894066022, 1.96468843076292419000102524742, 2.78430071245701437646230321943, 4.11847589579252634565171609498, 5.58112899469057236116017202583, 6.36265800113242475495925434693, 7.20170600090058637441302199710, 8.514371779755069026471824514854, 8.790593422885529292387892182084, 9.589162200564090696449566399648