| L(s) = 1 | + (0.0594 + 0.0745i)2-s + (0.443 − 1.94i)4-s + (0.284 − 3.80i)5-s + (1.30 + 2.26i)7-s + (0.342 − 0.165i)8-s + (0.300 − 0.204i)10-s + (0.456 + 1.99i)11-s + (−1.76 − 1.20i)13-s + (−0.0912 + 0.232i)14-s + (−3.55 − 1.71i)16-s + (−0.388 − 5.18i)17-s + (−0.603 − 0.559i)19-s + (−7.25 − 2.23i)20-s + (−0.121 + 0.152i)22-s + (2.60 + 0.805i)23-s + ⋯ |
| L(s) = 1 | + (0.0420 + 0.0527i)2-s + (0.221 − 0.970i)4-s + (0.127 − 1.70i)5-s + (0.495 + 0.857i)7-s + (0.121 − 0.0583i)8-s + (0.0950 − 0.0647i)10-s + (0.137 + 0.602i)11-s + (−0.490 − 0.334i)13-s + (−0.0243 + 0.0621i)14-s + (−0.888 − 0.427i)16-s + (−0.0942 − 1.25i)17-s + (−0.138 − 0.128i)19-s + (−1.62 − 0.500i)20-s + (−0.0260 + 0.0326i)22-s + (0.544 + 0.167i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0193 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0193 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05716 - 1.03687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05716 - 1.03687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-3.37 - 5.62i)T \) |
| good | 2 | \( 1 + (-0.0594 - 0.0745i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.284 + 3.80i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 2.26i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.456 - 1.99i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (1.76 + 1.20i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.388 + 5.18i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (0.603 + 0.559i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.60 - 0.805i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (1.43 - 3.65i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-7.40 + 1.11i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (0.431 - 0.746i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.40 - 3.02i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.00195 + 0.00857i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (3.12 - 2.13i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-3.54 - 1.70i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-6.20 - 0.935i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (9.35 + 8.67i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.220 + 0.0680i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-9.06 - 6.18i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.133 + 0.231i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.09 - 7.88i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (5.18 + 13.2i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.14 - 5.00i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−11.27949852216498799265866214529, −9.903212919392061378802019483234, −9.314511667318257696635793381276, −8.562921208032133328641711596308, −7.38545031026124059126673966442, −6.03485257468263977336343750700, −4.97676180339285968012016462918, −4.78704902237750127034705391413, −2.35261235712075248108937155057, −1.04263966701946857398866251092,
2.27321189455954705252345520428, 3.40341497237102412477520102139, 4.26537857769393277509634889688, 6.13634564213307344473725824490, 6.96166093937470598011942612722, 7.65024783170550745741078345826, 8.568372098003513063210208896362, 10.03926551086211007323635327699, 10.82644481127590424254529861693, 11.31213597692583178056528758205
