| L(s) = 1 | + (−0.486 − 2.00i)2-s + (0.327 + 0.945i)3-s + (−2.00 + 1.03i)4-s + (−3.02 + 1.21i)5-s + (1.73 − 1.11i)6-s + (1.86 + 1.87i)7-s + (0.342 + 0.395i)8-s + (−0.786 + 0.618i)9-s + (3.90 + 5.47i)10-s + (0.776 − 3.20i)11-s + (−1.63 − 1.55i)12-s + (2.65 − 5.81i)13-s + (2.86 − 4.64i)14-s + (−2.13 − 2.46i)15-s + (−1.98 + 2.79i)16-s + (−0.326 − 6.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.343 − 1.41i)2-s + (0.188 + 0.545i)3-s + (−1.00 + 0.516i)4-s + (−1.35 + 0.541i)5-s + (0.708 − 0.455i)6-s + (0.703 + 0.710i)7-s + (0.121 + 0.139i)8-s + (−0.262 + 0.206i)9-s + (1.23 + 1.73i)10-s + (0.234 − 0.964i)11-s + (−0.470 − 0.448i)12-s + (0.737 − 1.61i)13-s + (0.764 − 1.24i)14-s + (−0.551 − 0.636i)15-s + (−0.497 + 0.698i)16-s + (−0.0790 − 1.66i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.340491 - 0.819345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.340491 - 0.819345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.327 - 0.945i)T \) |
| 7 | \( 1 + (-1.86 - 1.87i)T \) |
| 23 | \( 1 + (-0.225 - 4.79i)T \) |
| good | 2 | \( 1 + (0.486 + 2.00i)T + (-1.77 + 0.916i)T^{2} \) |
| 5 | \( 1 + (3.02 - 1.21i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (-0.776 + 3.20i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (-2.65 + 5.81i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.326 + 6.84i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.248 + 5.21i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 0.680i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.11 + 0.600i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-4.37 + 3.44i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.196 + 1.36i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.12 - 2.44i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.84 - 0.557i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (0.320 + 0.449i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (-1.97 + 5.72i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (6.34 - 6.05i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (3.17 + 0.933i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (8.52 - 4.39i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (16.5 - 1.57i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (0.164 + 1.14i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.84 + 1.51i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (-2.47 + 17.1i)T + (-93.0 - 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00379494243994278106452866727, −9.983364162140301602009864055505, −8.917077461325495481139755376916, −8.343481136177961383537614430058, −7.33262345495505757659047020562, −5.67876430539431225579431773140, −4.43864812022012092112978514532, −3.23422235430369469362065856826, −2.84212689941462050341372302509, −0.64587888077746684968420221552,
1.52408235827630558468370657052, 4.04848780869281641231393257571, 4.54160605574593092684369725745, 6.16447775819339408885906542417, 6.92078958939533574770883095694, 7.74282613245588671929245574159, 8.321476923235704368253642430810, 8.875523098102629164127466827249, 10.30663125633786171670098352997, 11.56700807735232307579148422132
