L(s) = 1 | + (0.322 − 0.558i)2-s + (−1.15 + 1.28i)3-s + (0.792 + 1.37i)4-s + (−0.550 + 2.16i)5-s + (0.346 + 1.06i)6-s + (−2.64 + 0.105i)7-s + 2.31·8-s + (−0.319 − 2.98i)9-s + (1.03 + 1.00i)10-s + (3.51 − 2.02i)11-s + (−2.68 − 0.567i)12-s + 4.21·13-s + (−0.793 + 1.51i)14-s + (−2.15 − 3.21i)15-s + (−0.839 + 1.45i)16-s + (1.88 − 1.08i)17-s + ⋯ |
L(s) = 1 | + (0.227 − 0.394i)2-s + (−0.668 + 0.743i)3-s + (0.396 + 0.685i)4-s + (−0.246 + 0.969i)5-s + (0.141 + 0.433i)6-s + (−0.999 + 0.0397i)7-s + 0.817·8-s + (−0.106 − 0.994i)9-s + (0.326 + 0.318i)10-s + (1.05 − 0.611i)11-s + (−0.774 − 0.163i)12-s + 1.16·13-s + (−0.212 + 0.403i)14-s + (−0.556 − 0.830i)15-s + (−0.209 + 0.363i)16-s + (0.457 − 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.850646 + 0.479642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850646 + 0.479642i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.15 - 1.28i)T \) |
| 5 | \( 1 + (0.550 - 2.16i)T \) |
| 7 | \( 1 + (2.64 - 0.105i)T \) |
good | 2 | \( 1 + (-0.322 + 0.558i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-3.51 + 2.02i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + (-1.88 + 1.08i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.87 + 2.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.322 - 0.558i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.16iT - 29T^{2} \) |
| 31 | \( 1 + (0.339 - 0.195i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.69 - 2.13i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 + 6.54iT - 43T^{2} \) |
| 47 | \( 1 + (6.75 + 3.90i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.60 + 6.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.66 + 9.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.05 - 3.49i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.56 - 4.36i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.13iT - 71T^{2} \) |
| 73 | \( 1 + (2.61 + 4.53i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.87 - 3.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.27iT - 83T^{2} \) |
| 89 | \( 1 + (0.447 - 0.774i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−13.85028814861969750882002884277, −12.67773869471600451376523390110, −11.55989913551281207024739939031, −11.06317439046278664099960305540, −10.00426470295355964489685876850, −8.654775394482522614803922911242, −6.87039414161545320405319029065, −6.15757224532687528355769937967, −3.95989342481164444408658015231, −3.22454018178035205889359300612,
1.39281036532228703720151054819, 4.33402355750316525292965349509, 5.90943191946909424416573825390, 6.47113962505126988678222195331, 7.81124731789452664396494128916, 9.282811194284658584105336219906, 10.54288030495951722297130835233, 11.70196332433311297692512836622, 12.65270840319415574697817888160, 13.40900658602365553342313966600