| 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
−9.921842364377152059602527930488, −9.011622966704414749863295463424, −8.102086162716399796020974519679, −7.62532073822940633844813252572, −6.30984728036640776482711777431, −5.33144718447756177406318804220, −5.25111182673596992396015425864, −3.90418615405526586498808338841, −2.48739672640388114517042932277, −0.58766490202990709272815771990,
1.60762186808791676071564877858, 2.34672924855340673485954841031, 4.16781678959212607907995045283, 4.63075209655663142380646405306, 5.59680897457264314635230812967, 6.96150324909142487974592745629, 7.29781818108543650538725314653, 8.630099632102814430975304661033, 9.604003256976822925326773998276, 10.35164955085104215111669828019
