L(s) = 1 | + (0.309 − 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (−0.258 + 2.45i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (2.63 − 1.17i)11-s + (−0.978 + 0.207i)12-s + (1.33 + 0.283i)13-s + (2.25 + 1.00i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (4.71 + 2.09i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (0.386 − 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.0976 + 0.928i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (0.211 + 0.235i)10-s + (0.793 − 0.353i)11-s + (−0.282 + 0.0600i)12-s + (0.369 + 0.0786i)13-s + (0.603 + 0.268i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (1.14 + 0.508i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.80948 - 0.842133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80948 - 0.842133i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-1.22 + 5.43i)T \) |
good | 7 | \( 1 + (0.258 - 2.45i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.63 + 1.17i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 0.283i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-4.71 - 2.09i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.78 + 0.378i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 1.35i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.38 + 4.26i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.161 + 0.280i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.77 - 6.41i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.643 + 0.136i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-3.34 - 10.3i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.899 + 8.55i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (3.56 - 3.95i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 + (-2.04 + 3.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0859 - 0.818i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.63 - 2.95i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (7.67 + 3.41i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.86 - 2.06i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (3.13 + 2.27i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.53 - 4.74i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.858411969757189671115998078873, −9.213833513386456364837301724029, −8.373046051990567075885200046954, −7.57427360128537374513020896821, −6.28916654138665671893198347336, −5.78023091235902938982823749507, −4.34862624247418264551333090740, −3.33924677477962436234639726240, −2.53183461715728707158674498943, −1.19065184156262634452545315550,
1.15102812606467828183958100704, 3.21369758857264527556455867273, 3.96406122801695600426589554649, 4.84791003448494485237256231592, 5.76791587025613000244089713444, 7.04571507994849736778561676088, 7.47188060497444303683253075971, 8.532792748470567813204985193080, 9.213245409706995750751483661957, 10.03077767046116501093107081298