| 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 + (−2.05 + 0.914i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.0978 + 0.930i)11-s + (0.669 + 0.743i)12-s + (−1.58 + 1.75i)13-s + (0.234 + 2.23i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.108 − 1.02i)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 + (−0.775 + 0.345i)7-s + (−0.286 + 0.207i)8-s + (0.304 + 0.135i)9-s + (−0.309 + 0.0657i)10-s + (−0.0294 + 0.280i)11-s + (0.193 + 0.214i)12-s + (−0.439 + 0.488i)13-s + (0.0627 + 0.597i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.0262 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.923667 + 0.119833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.923667 + 0.119833i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5.38 + 1.43i)T \) |
| good | 7 | \( 1 + (2.05 - 0.914i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.0978 - 0.930i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.58 - 1.75i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.108 + 1.02i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-4.82 - 5.35i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.80 + 2.04i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.38 - 4.26i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (1.47 - 2.55i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.336 - 0.0716i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.815 - 0.905i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.920 - 2.83i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.57 - 2.48i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 2.39i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 5.29T + 61T^{2} \) |
| 67 | \( 1 + (1.21 + 2.09i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.31 - 2.36i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (1.06 - 10.1i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.368 - 3.50i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-9.52 + 2.02i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (1.59 + 1.15i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (13.1 + 9.58i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.02816171771684850978672759129, −9.563876736552173889705234432529, −8.619806695109529555206615430290, −7.50625948103424110070811998374, −6.57071448047901454909557173247, −5.59267057171063443827582068845, −4.81043335804716462667504898689, −3.75968516073985152417969450591, −2.63598834559812277091574040606, −1.18403338741158633658198134952,
0.51101202776635376010696656568, 2.86457393058742096174809291048, 3.79000528759946195608928717813, 4.93505195131779062191583907360, 5.71733770474474338733188024177, 6.72503705992821977277308936515, 7.20321585120688309095105408513, 8.163034249638055754489212806646, 9.301495209736045139378525757525, 9.956527167403320530990275078640
