| L(s) = 1 | + (−1.33 + 0.855i)2-s + (0.989 + 0.142i)3-s + (0.210 − 0.460i)4-s + (4.17 − 1.22i)5-s + (−1.44 + 0.657i)6-s + (−2.59 + 0.530i)7-s + (−0.336 − 2.34i)8-s + (0.959 + 0.281i)9-s + (−4.50 + 5.20i)10-s + (−3.50 + 5.44i)11-s + (0.273 − 0.425i)12-s + (2.11 + 1.83i)13-s + (2.99 − 2.92i)14-s + (4.30 − 0.618i)15-s + (3.11 + 3.59i)16-s + (1.32 + 2.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.941 + 0.605i)2-s + (0.571 + 0.0821i)3-s + (0.105 − 0.230i)4-s + (1.86 − 0.547i)5-s + (−0.587 + 0.268i)6-s + (−0.979 + 0.200i)7-s + (−0.118 − 0.827i)8-s + (0.319 + 0.0939i)9-s + (−1.42 + 1.64i)10-s + (−1.05 + 1.64i)11-s + (0.0790 − 0.122i)12-s + (0.586 + 0.508i)13-s + (0.801 − 0.781i)14-s + (1.11 − 0.159i)15-s + (0.778 + 0.898i)16-s + (0.322 + 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.949580 + 0.786082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949580 + 0.786082i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.59 - 0.530i)T \) |
| 23 | \( 1 + (-3.53 - 3.23i)T \) |
| good | 2 | \( 1 + (1.33 - 0.855i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (-4.17 + 1.22i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (3.50 - 5.44i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.11 - 1.83i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 2.90i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.41 + 3.09i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.356 + 0.780i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.03 + 0.148i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.117 - 0.400i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.55 - 5.28i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.78 - 0.831i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 0.395iT - 47T^{2} \) |
| 53 | \( 1 + (1.42 - 1.23i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (3.12 + 2.70i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.75 + 12.1i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.36 + 5.22i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (4.63 - 2.97i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (7.75 + 3.54i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (10.8 + 9.43i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (13.0 + 3.83i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.0377 - 0.262i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-8.56 + 2.51i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.49420807018768640636274768289, −9.820369269119329355153901309762, −9.391021737283930642001943020035, −8.797258573396214026796024087940, −7.60970671704629268371087572334, −6.70875148899958037895489610517, −5.85657824115849678639361454749, −4.63597271065062930648057369293, −2.90503545863186146027704415527, −1.63331277078164760003492220974,
1.05288092247671767548868902934, 2.66035703679589819490819482272, 3.05692688673978161426689692591, 5.56631605178948347824359193471, 5.93326282165351981040284004474, 7.27183058558183849465626906364, 8.569423705967619563740368645363, 9.111405691861935808968158945040, 10.05268965537591998306080005446, 10.40054046516944983223610437756
