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
−10.44328410486602634514917582168, −9.870340791477472875473698114491, −9.330648818905654315001142949100, −8.188060003342537695046214272329, −6.78865122783691683369462154614, −6.48052936331648729484335231281, −4.98108921154046475862918204653, −4.11590632582767566729608249224, −2.68845310042261494810598392595, −1.05533320838103773268359733883,
1.96310245565956689671840601858, 3.49145603236746502883473004765, 3.86728213138529580007700647160, 5.64547609670860024053496025857, 6.33235269213383121051403291329, 7.80569856518492187099263865121, 8.611541398579934103014160531086, 9.133265425327233321892015194198, 10.03066283637242337070542242637, 11.31025147212167082528710853357