| 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 + (−0.894 − 2.48i)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 + (−0.939 + 4.08i)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 + (−0.338 − 0.941i)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 + (−0.251 + 1.09i)14-s + (1.11 + 0.159i)15-s + (0.778 − 0.898i)16-s + (−0.322 + 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00966955 + 0.00372143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00966955 + 0.00372143i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.894 + 2.48i)T \) |
| 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
−10.94965690633134687103899100685, −10.61539208461015701110183947085, −9.264637439179918514372261935226, −8.393488134515715037403164674564, −7.82234554304353876150260922132, −6.73742100971279255465534131556, −5.19984173284504040818108774370, −4.27296357948473921786190429428, −3.08155235887714947540607114710, −0.76760092122562886166423119494,
0.01460086410095746568658921028, 2.78308557897335438860558652271, 4.14338128841169277252023498851, 5.26390026769808224870234990924, 6.76582665550019171844451636429, 7.43338461101304712248743988611, 7.84910548580338081624114920114, 8.936090050103635146417965384472, 9.959107318413516791166187387690, 10.72425279539181907928670371349
