L(s) = 1 | + (−0.309 − 0.535i)2-s + (0.809 − 1.40i)4-s − 5-s + (−2.11 + 3.66i)7-s − 2.23·8-s + (0.309 + 0.535i)10-s + (−0.118 − 0.204i)11-s + (−1 + 3.46i)13-s + 2.61·14-s + (−0.927 − 1.60i)16-s + (−1.73 + 3.00i)17-s + (−2.11 + 3.66i)19-s + (−0.809 + 1.40i)20-s + (−0.0729 + 0.126i)22-s + (1.88 + 3.25i)23-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.378i)2-s + (0.404 − 0.700i)4-s − 0.447·5-s + (−0.800 + 1.38i)7-s − 0.790·8-s + (0.0977 + 0.169i)10-s + (−0.0355 − 0.0616i)11-s + (−0.277 + 0.960i)13-s + 0.699·14-s + (−0.231 − 0.401i)16-s + (−0.421 + 0.729i)17-s + (−0.485 + 0.841i)19-s + (−0.180 + 0.313i)20-s + (−0.0155 + 0.0269i)22-s + (0.392 + 0.679i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.437999 + 0.432418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.437999 + 0.432418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 + (0.309 + 0.535i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (2.11 - 3.66i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.118 + 0.204i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.73 - 3.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.11 - 3.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 - 3.25i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.73 + 6.47i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.97 - 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.11 + 5.40i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (0.354 - 0.613i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.20 - 12.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.35 - 2.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.11 - 5.40i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 + 3.00i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00735536856705118305422813292, −9.951864488539769568573126607849, −9.279759827797516489549651628550, −8.588809140134311489226908721013, −7.27855243821300389808197538421, −6.15771796462568575908585185226, −5.73971289702522364990185617485, −4.22578025241589651508972337880, −2.85824117381228768039215392628, −1.85338986899159605204225231246,
0.34666713397702527086615944668, 2.78320810426346211461340491440, 3.64664699179982748449025053662, 4.75138525560885738486016018059, 6.28459494911897342099917778355, 7.18577817949403755617788900360, 7.48672294730012299086966117904, 8.634127983199081182706011819775, 9.490267378032233923526017124576, 10.70323563179555722153570479996