| L(s) = 1 | + (−0.224 − 1.39i)2-s + (0.283 + 1.70i)3-s + (−1.89 + 0.627i)4-s + (0.747 − 2.05i)5-s + (2.32 − 0.779i)6-s + (−0.984 − 0.173i)7-s + (1.30 + 2.51i)8-s + (−2.83 + 0.967i)9-s + (−3.03 − 0.581i)10-s + (−3.57 + 1.30i)11-s + (−1.61 − 3.06i)12-s + (0.135 − 0.113i)13-s + (−0.0209 + 1.41i)14-s + (3.72 + 0.695i)15-s + (3.21 − 2.38i)16-s + (−5.31 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.159 − 0.987i)2-s + (0.163 + 0.986i)3-s + (−0.949 + 0.313i)4-s + (0.334 − 0.918i)5-s + (0.947 − 0.318i)6-s + (−0.372 − 0.0656i)7-s + (0.460 + 0.887i)8-s + (−0.946 + 0.322i)9-s + (−0.960 − 0.184i)10-s + (−1.07 + 0.392i)11-s + (−0.464 − 0.885i)12-s + (0.0375 − 0.0314i)13-s + (−0.00561 + 0.377i)14-s + (0.960 + 0.179i)15-s + (0.802 − 0.596i)16-s + (−1.28 + 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.199867 + 0.303108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.199867 + 0.303108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.224 + 1.39i)T \) |
| 3 | \( 1 + (-0.283 - 1.70i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-0.747 + 2.05i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.57 - 1.30i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.135 + 0.113i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.31 - 3.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.99 - 1.15i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0492 + 0.279i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.88 - 5.81i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.41 - 0.777i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.564 - 0.977i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.24 + 8.63i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.101 + 0.279i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.47 - 8.37i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 + (-3.05 - 1.11i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.13 - 12.1i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.21 + 5.01i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0796 + 0.137i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.16 - 7.21i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.06 + 1.27i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.85 + 5.75i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.26 - 0.728i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 + 4.42i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.54053191985784302948289972948, −9.861441786588505446456010970282, −8.957372837543901776028998707085, −8.658812805903655242629392629952, −7.45741466633316437436926709571, −5.68015535303176317181924718067, −4.99467771901922963282412384705, −4.14490714123183172825609810962, −3.09302873276781192371796663994, −1.86417015110102096649453340268,
0.17947760681582571721125122335, 2.26300596988045912832067467880, 3.35741786116894881279230880621, 5.01848444540401807416419409606, 5.97770205943034395469168986783, 6.68150220375064836669520710927, 7.32785300432690396653218424003, 8.123388342284646825275261361515, 9.028277706672940448089973543687, 9.867581391505071638717369953909
