| L(s) = 1 | + (1.71 + 1.63i)2-s + (−0.981 + 0.189i)3-s + (0.170 + 3.58i)4-s + (2.82 − 2.22i)5-s + (−1.98 − 1.27i)6-s + (−1.76 + 1.97i)7-s + (−2.46 + 2.84i)8-s + (0.928 − 0.371i)9-s + (8.45 + 0.807i)10-s + (−3.32 + 3.17i)11-s + (−0.846 − 3.48i)12-s + (2.20 + 4.82i)13-s + (−6.23 + 0.501i)14-s + (−2.35 + 2.71i)15-s + (−1.69 + 0.162i)16-s + (1.26 − 0.650i)17-s + ⋯ |
| L(s) = 1 | + (1.20 + 1.15i)2-s + (−0.566 + 0.109i)3-s + (0.0854 + 1.79i)4-s + (1.26 − 0.993i)5-s + (−0.811 − 0.521i)6-s + (−0.666 + 0.745i)7-s + (−0.870 + 1.00i)8-s + (0.309 − 0.123i)9-s + (2.67 + 0.255i)10-s + (−1.00 + 0.956i)11-s + (−0.244 − 1.00i)12-s + (0.610 + 1.33i)13-s + (−1.66 + 0.133i)14-s + (−0.607 + 0.701i)15-s + (−0.424 + 0.0405i)16-s + (0.306 − 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43236 + 2.05467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43236 + 2.05467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.981 - 0.189i)T \) |
| 7 | \( 1 + (1.76 - 1.97i)T \) |
| 23 | \( 1 + (-1.87 + 4.41i)T \) |
| good | 2 | \( 1 + (-1.71 - 1.63i)T + (0.0951 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-2.82 + 2.22i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (3.32 - 3.17i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.20 - 4.82i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 0.650i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-4.64 - 2.39i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (4.42 + 2.84i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.56 + 4.51i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (-0.228 + 0.0913i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (0.958 + 6.66i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.20 + 8.31i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (3.98 + 6.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.49 + 3.50i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (0.392 + 0.0374i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 2.18i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (3.06 - 12.6i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 3.33i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.447 + 9.38i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (4.84 - 6.80i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (0.423 - 2.94i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (1.06 - 3.06i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-0.806 - 5.61i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.77450278815882408997032750148, −10.11116233377847543863422253969, −9.461119447324653710596220372225, −8.457547086221631997255119999397, −7.12972076963748132971386615675, −6.33490008406106074772113882232, −5.40653903457858137916239895595, −5.17082905171960567733890977159, −3.90815369515415731012338261724, −2.11934106935043439866903402781,
1.28001076760769220417214681157, 3.10045894937948129788710810048, 3.20755239737716402947836794297, 5.16441185378035546977165570814, 5.69607908001846884408288774871, 6.51430354546835442779031967713, 7.74887710845266571754328626266, 9.675044854407626632155558263943, 10.19784982248719090676287211972, 10.99464586189482957179396000601
