| L(s) = 1 | + (1.50 − 2.11i)2-s + (0.235 − 0.971i)3-s + (−1.55 − 4.48i)4-s + (0.00580 − 0.121i)5-s + (−1.70 − 1.96i)6-s + (−0.123 − 2.64i)7-s + (−6.84 − 2.01i)8-s + (−0.888 − 0.458i)9-s + (−0.249 − 0.195i)10-s + (2.37 + 3.33i)11-s + (−4.72 + 0.451i)12-s + (−0.538 + 3.74i)13-s + (−5.77 − 3.72i)14-s + (−0.117 − 0.0343i)15-s + (−7.11 + 5.59i)16-s + (4.04 − 0.780i)17-s + ⋯ |
| L(s) = 1 | + (1.06 − 1.49i)2-s + (0.136 − 0.561i)3-s + (−0.776 − 2.24i)4-s + (0.00259 − 0.0545i)5-s + (−0.694 − 0.801i)6-s + (−0.0466 − 0.998i)7-s + (−2.42 − 0.711i)8-s + (−0.296 − 0.152i)9-s + (−0.0788 − 0.0619i)10-s + (0.716 + 1.00i)11-s + (−1.36 + 0.130i)12-s + (−0.149 + 1.03i)13-s + (−1.54 − 0.994i)14-s + (−0.0302 − 0.00887i)15-s + (−1.77 + 1.39i)16-s + (0.981 − 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0639267 + 2.38440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0639267 + 2.38440i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (0.123 + 2.64i)T \) |
| 23 | \( 1 + (4.79 + 0.0569i)T \) |
| good | 2 | \( 1 + (-1.50 + 2.11i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.00580 + 0.121i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 3.33i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.538 - 3.74i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.04 + 0.780i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-2.95 - 0.570i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (5.53 + 6.38i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.426 - 0.406i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (9.29 + 4.79i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-5.96 - 3.83i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.93 + 0.567i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-1.92 - 3.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.89 - 2.36i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-9.29 - 7.30i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (2.53 + 10.4i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (1.09 + 0.104i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (-2.60 - 5.69i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.57 - 7.44i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-5.30 + 2.12i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-3.56 + 2.29i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 10.7i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (8.05 + 5.17i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.77005650974422711183020661767, −9.833334870278959680381774945061, −9.295352057240263003309370732429, −7.61130665581792250517828721885, −6.70272255648782699843621533884, −5.47907630041279061044639960344, −4.30250826450083962841382819622, −3.66583624240237414351247336523, −2.21087734271048591132169408411, −1.16555728358050256721320090271,
3.08532445640090498740382200842, 3.75414575889930073078502114892, 5.32961167348342676435893329124, 5.51444508433272462675472023728, 6.58975053843474385528345794877, 7.73549743844923046161284909437, 8.531064912455116934013874681950, 9.212260851036098402778841436540, 10.54610064732945113829385282800, 11.84895817554747599919816074322
