L(s) = 1 | + (1.43 − 2.48i)3-s − 3.54·7-s + (−2.62 − 4.54i)9-s − 1.81·11-s + (1.60 + 2.78i)13-s + (−3.99 + 6.92i)17-s + (−0.863 + 4.27i)19-s + (−5.09 + 8.82i)21-s + (−4.21 − 7.30i)23-s − 6.46·27-s + (4.29 + 7.43i)29-s − 1.70·31-s + (−2.60 + 4.51i)33-s − 5.50·37-s + 9.23·39-s + ⋯ |
L(s) = 1 | + (0.829 − 1.43i)3-s − 1.34·7-s + (−0.875 − 1.51i)9-s − 0.547·11-s + (0.445 + 0.771i)13-s + (−0.969 + 1.67i)17-s + (−0.198 + 0.980i)19-s + (−1.11 + 1.92i)21-s + (−0.878 − 1.52i)23-s − 1.24·27-s + (0.796 + 1.38i)29-s − 0.306·31-s + (−0.453 + 0.786i)33-s − 0.905·37-s + 1.47·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0133 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0133 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3717494356\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3717494356\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.863 - 4.27i)T \) |
good | 3 | \( 1 + (-1.43 + 2.48i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + (-1.60 - 2.78i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.99 - 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.21 + 7.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.29 - 7.43i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 + (-4.05 + 7.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.51 - 4.35i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.674 + 1.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.10 + 1.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.960 - 1.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.83 - 4.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.64 + 8.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.94 - 5.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.63 + 2.82i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.08 - 3.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.30T + 83T^{2} \) |
| 89 | \( 1 + (2.73 + 4.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.99 + 6.91i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.025684782693828517009198402114, −8.579339395352379937597656565894, −7.952633411666204636706273293306, −6.83442534366128022592779192241, −6.54555469135610787873556310653, −5.85111370720168866857992001279, −4.19150494320977574801906767216, −3.36724367374090822779954432075, −2.38057181550254575973775279441, −1.56771897758567467692143218410,
0.11158201883957409702420438940, 2.57396455731218973048031639074, 3.05909162519151272202847045645, 3.92235025837724925695358501584, 4.79569882909819742905471852405, 5.61695882964196553178670331373, 6.63930842089068886132530038936, 7.60578136918526779827729309045, 8.471974980118081478421881731796, 9.279489480809386431280127913519