L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (2.69 − 1.19i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (−0.511 + 4.86i)11-s + (0.669 + 0.743i)12-s + (−3.88 + 4.31i)13-s + (−0.308 − 2.93i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.454 + 4.32i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.564 − 0.120i)3-s + (−0.404 − 0.293i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (1.01 − 0.453i)7-s + (−0.286 + 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (−0.154 + 1.46i)11-s + (0.193 + 0.214i)12-s + (−1.07 + 1.19i)13-s + (−0.0824 − 0.784i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.110 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17041 + 0.391580i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17041 + 0.391580i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (4.43 - 3.36i)T \) |
good | 7 | \( 1 + (-2.69 + 1.19i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.511 - 4.86i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (3.88 - 4.31i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.454 - 4.32i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (2.85 + 3.16i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.665 + 0.483i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.37 - 4.22i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.22 + 2.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.37 + 0.716i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.18 - 4.64i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (1.31 + 4.04i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 4.69i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-11.3 - 2.41i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + (-7.38 - 12.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.97 + 2.21i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.565 - 5.37i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.418 + 3.98i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-12.6 + 2.68i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (13.2 + 9.65i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (15.0 + 10.9i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.35164955085104215111669828019, −9.604003256976822925326773998276, −8.630099632102814430975304661033, −7.29781818108543650538725314653, −6.96150324909142487974592745629, −5.59680897457264314635230812967, −4.63075209655663142380646405306, −4.16781678959212607907995045283, −2.34672924855340673485954841031, −1.60762186808791676071564877858,
0.58766490202990709272815771990, 2.48739672640388114517042932277, 3.90418615405526586498808338841, 5.25111182673596992396015425864, 5.33144718447756177406318804220, 6.30984728036640776482711777431, 7.62532073822940633844813252572, 8.102086162716399796020974519679, 9.011622966704414749863295463424, 9.921842364377152059602527930488