L(s) = 1 | + (5.53 − 1.14i)2-s + 16.0·3-s + (29.3 − 12.7i)4-s + (−19.1 − 52.5i)5-s + (88.6 − 18.3i)6-s + 20.5i·7-s + (148. − 104. i)8-s + 13.2·9-s + (−166. − 269. i)10-s + 619. i·11-s + (470. − 203. i)12-s + 101.·13-s + (23.5 + 113. i)14-s + (−306. − 840. i)15-s + (701. − 746. i)16-s − 527. i·17-s + ⋯ |
L(s) = 1 | + (0.979 − 0.202i)2-s + 1.02·3-s + (0.917 − 0.397i)4-s + (−0.342 − 0.939i)5-s + (1.00 − 0.208i)6-s + 0.158i·7-s + (0.818 − 0.574i)8-s + 0.0545·9-s + (−0.525 − 0.850i)10-s + 1.54i·11-s + (0.942 − 0.407i)12-s + 0.166·13-s + (0.0321 + 0.155i)14-s + (−0.351 − 0.964i)15-s + (0.684 − 0.728i)16-s − 0.442i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.33400 - 1.04823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.33400 - 1.04823i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.53 + 1.14i)T \) |
| 5 | \( 1 + (19.1 + 52.5i)T \) |
good | 3 | \( 1 - 16.0T + 243T^{2} \) |
| 7 | \( 1 - 20.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 619. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 101.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 527. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.54e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.30e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 201.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.62e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.99e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.03e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 7.81e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.04e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.94e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.59e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.35e5iT - 8.58e9T^{2} \) |
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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
−14.86706554030239722359500415890, −13.88774917773402672737924060990, −12.69246925543831202996740119684, −11.93530481430421453684644542778, −10.08187060446637298520525016069, −8.662200525676068393281182111191, −7.26271846993738680747004595924, −5.19313855203525005134447079195, −3.78162315712296714617613790941, −1.97238719627609302848738332908,
2.73368435806764485876573801211, 3.73903288597570934770096020120, 5.97977203971889078349933629390, 7.44198332537150628189003500267, 8.632480604812951655072762697141, 10.72062681522203789699849365723, 11.67013707468122287761333585042, 13.50456977392235338499310894269, 13.96535257144153466460969270653, 15.05551732146880146967603304966