| L(s) = 1 | + (0.227 − 0.146i)2-s + (0.989 + 0.142i)3-s + (−0.800 + 1.75i)4-s + (1.03 − 0.305i)5-s + (0.245 − 0.112i)6-s + (−1.80 + 1.93i)7-s + (0.150 + 1.04i)8-s + (0.959 + 0.281i)9-s + (0.191 − 0.221i)10-s + (2.42 − 3.78i)11-s + (−1.04 + 1.62i)12-s + (2.38 + 2.06i)13-s + (−0.126 + 0.703i)14-s + (1.07 − 0.154i)15-s + (−2.33 − 2.69i)16-s + (2.18 + 4.78i)17-s + ⋯ |
| L(s) = 1 | + (0.160 − 0.103i)2-s + (0.571 + 0.0821i)3-s + (−0.400 + 0.876i)4-s + (0.465 − 0.136i)5-s + (0.100 − 0.0458i)6-s + (−0.680 + 0.732i)7-s + (0.0533 + 0.371i)8-s + (0.319 + 0.0939i)9-s + (0.0606 − 0.0699i)10-s + (0.732 − 1.13i)11-s + (−0.300 + 0.467i)12-s + (0.660 + 0.572i)13-s + (−0.0337 + 0.187i)14-s + (0.277 − 0.0398i)15-s + (−0.584 − 0.674i)16-s + (0.530 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48129 + 0.923852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48129 + 0.923852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 - 0.142i)T \) |
| 7 | \( 1 + (1.80 - 1.93i)T \) |
| 23 | \( 1 + (0.277 - 4.78i)T \) |
| good | 2 | \( 1 + (-0.227 + 0.146i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.03 + 0.305i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 3.78i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.38 - 2.06i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 4.78i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.28 - 5.01i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.778 + 1.70i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-6.66 + 0.957i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.08 + 7.10i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.11 + 10.6i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.39 + 0.200i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 1.11iT - 47T^{2} \) |
| 53 | \( 1 + (1.39 - 1.20i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-9.62 - 8.34i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.66 + 11.5i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (5.58 + 8.68i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-8.57 + 5.51i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-12.5 - 5.74i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.67 + 5.77i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.695 - 0.204i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.66 + 11.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-7.69 + 2.25i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.31025147212167082528710853357, −10.03066283637242337070542242637, −9.133265425327233321892015194198, −8.611541398579934103014160531086, −7.80569856518492187099263865121, −6.33235269213383121051403291329, −5.64547609670860024053496025857, −3.86728213138529580007700647160, −3.49145603236746502883473004765, −1.96310245565956689671840601858,
1.05533320838103773268359733883, 2.68845310042261494810598392595, 4.11590632582767566729608249224, 4.98108921154046475862918204653, 6.48052936331648729484335231281, 6.78865122783691683369462154614, 8.188060003342537695046214272329, 9.330648818905654315001142949100, 9.870340791477472875473698114491, 10.44328410486602634514917582168
