| L(s) = 1 | + (−2.07 + 1.33i)2-s + (0.989 + 0.142i)3-s + (1.68 − 3.69i)4-s + (−3.92 + 1.15i)5-s + (−2.23 + 1.02i)6-s + (−2.13 − 1.55i)7-s + (0.723 + 5.03i)8-s + (0.959 + 0.281i)9-s + (6.59 − 7.60i)10-s + (1.89 − 2.94i)11-s + (2.19 − 3.41i)12-s + (−0.793 − 0.687i)13-s + (6.50 + 0.379i)14-s + (−4.04 + 0.581i)15-s + (−2.87 − 3.31i)16-s + (2.66 + 5.83i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 + 0.941i)2-s + (0.571 + 0.0821i)3-s + (0.844 − 1.84i)4-s + (−1.75 + 0.515i)5-s + (−0.914 + 0.417i)6-s + (−0.808 − 0.588i)7-s + (0.255 + 1.77i)8-s + (0.319 + 0.0939i)9-s + (2.08 − 2.40i)10-s + (0.570 − 0.887i)11-s + (0.634 − 0.986i)12-s + (−0.220 − 0.190i)13-s + (1.73 + 0.101i)14-s + (−1.04 + 0.150i)15-s + (−0.718 − 0.828i)16-s + (0.646 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.480064 + 0.213220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.480064 + 0.213220i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.13 + 1.55i)T \) |
| 23 | \( 1 + (-4.30 - 2.11i)T \) |
| good | 2 | \( 1 + (2.07 - 1.33i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (3.92 - 1.15i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-1.89 + 2.94i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.793 + 0.687i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.66 - 5.83i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.346 - 0.758i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.92 + 6.39i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-8.75 + 1.25i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.804 - 2.73i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.15 - 3.93i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-9.48 - 1.36i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 7.61iT - 47T^{2} \) |
| 53 | \( 1 + (0.894 - 0.775i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (6.11 + 5.30i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.390 - 2.71i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.02 - 4.70i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 7.65i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.76 + 1.26i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (2.98 + 2.58i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.67 - 1.07i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.689 - 4.79i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.41 + 1.00i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.75113042097011727709828696634, −10.05242682680080206122883062822, −9.076016714871770299815922984944, −8.096174661812178671709898043483, −7.82715165958075546127894551069, −6.86193933945337085541995266689, −6.11273637850958964257119685821, −4.08640889600630626890574286405, −3.22813684548877607286404053940, −0.76243911308514789703110933095,
0.845151627449695103470849642955, 2.64680555120858186227760436217, 3.50058763143199544106825603081, 4.71968316531363069804249229319, 7.04922132846216497780684285183, 7.45164786736673484705583977973, 8.491949812820611629392735696628, 9.135011051179233104789505578830, 9.646592697011498512008995514124, 10.86212575102829422408855186052
