L(s) = 1 | + (1.02 + 0.593i)2-s + (0.298 + 0.172i)3-s + (−0.295 − 0.511i)4-s + (−1.44 + 1.71i)5-s + (0.204 + 0.354i)6-s + (−1.75 + 1.01i)7-s − 3.07i·8-s + (−1.44 − 2.49i)9-s + (−2.49 + 0.903i)10-s + (1.94 − 3.36i)11-s − 0.203i·12-s − 2.40·14-s + (−0.725 + 0.262i)15-s + (1.23 − 2.14i)16-s + (−4.71 + 2.72i)17-s − 3.42i·18-s + ⋯ |
L(s) = 1 | + (0.727 + 0.419i)2-s + (0.172 + 0.0996i)3-s + (−0.147 − 0.255i)4-s + (−0.644 + 0.764i)5-s + (0.0836 + 0.144i)6-s + (−0.664 + 0.383i)7-s − 1.08i·8-s + (−0.480 − 0.831i)9-s + (−0.789 + 0.285i)10-s + (0.585 − 1.01i)11-s − 0.0588i·12-s − 0.644·14-s + (−0.187 + 0.0678i)15-s + (0.308 − 0.535i)16-s + (−1.14 + 0.660i)17-s − 0.806i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.615231 - 0.725073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615231 - 0.725073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.44 - 1.71i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.02 - 0.593i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.298 - 0.172i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.75 - 1.01i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.94 + 3.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4.71 - 2.72i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.298 + 0.172i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + (-4.71 - 2.72i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0902 + 0.156i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.15 + 0.669i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.81 + 2.20i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.940 + 1.62i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 8.86iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.83iT - 83T^{2} \) |
| 89 | \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.02 + 2.90i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.910729900101908228177653843534, −9.011975496165161406272413955942, −8.428104391927506380538262335565, −6.88062937014366250236581826101, −6.45577014794520577972446723622, −5.80650551088049552580800999673, −4.36988213415471209956802371597, −3.67420362109502054565241802692, −2.74294729018876873706773542249, −0.34469167441404592145031893089,
1.92659553306154573843315534524, 3.14082429397210806417979602773, 4.27079362576278521211880381014, 4.62914931692538913735133802423, 5.84557004810064783507099355609, 7.13746027016927973154363323548, 7.907530470094238894842404373906, 8.724670452234040497102791391075, 9.460803327494802251648278059756, 10.65117040418473649778875736932