L(s) = 1 | + (−2.40 − 1.49i)2-s + (−3.96 + 3.36i)3-s + (3.54 + 7.17i)4-s + (−9.10 − 6.48i)5-s + (14.5 − 2.16i)6-s + 26.5·7-s + (2.20 − 22.5i)8-s + (4.37 − 26.6i)9-s + (12.1 + 29.1i)10-s + 17.2·11-s + (−38.1 − 16.5i)12-s − 66.7i·13-s + (−63.7 − 39.6i)14-s + (57.8 − 4.95i)15-s + (−38.9 + 50.7i)16-s + 87.8·17-s + ⋯ |
L(s) = 1 | + (−0.849 − 0.527i)2-s + (−0.762 + 0.647i)3-s + (0.442 + 0.896i)4-s + (−0.814 − 0.579i)5-s + (0.989 − 0.147i)6-s + 1.43·7-s + (0.0976 − 0.995i)8-s + (0.162 − 0.986i)9-s + (0.385 + 0.922i)10-s + 0.471·11-s + (−0.917 − 0.397i)12-s − 1.42i·13-s + (−1.21 − 0.756i)14-s + (0.996 − 0.0852i)15-s + (−0.608 + 0.793i)16-s + 1.25·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.635162 - 0.358216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.635162 - 0.358216i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.40 + 1.49i)T \) |
| 3 | \( 1 + (3.96 - 3.36i)T \) |
| 5 | \( 1 + (9.10 + 6.48i)T \) |
good | 7 | \( 1 - 26.5T + 343T^{2} \) |
| 11 | \( 1 - 17.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 87.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 22.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 30.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 54.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 143. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 285. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 57.8iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 111. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 160.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 447.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 559.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 253. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 859. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 79.1iT - 9.12e5T^{2} \) |
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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
−14.81357312559052332852859650724, −12.69514292296110417421377362406, −11.73928208833212504807008782590, −11.09455401840389667513976536761, −9.916485678204595170028017423559, −8.508031113356670204841977091586, −7.56811236958841945271987991957, −5.30732308048382708556108505893, −3.81900264819751348242576029799, −0.866666451540655846143417814462,
1.49918794050993992341101863750, 4.86865499445972552918059147316, 6.52019005040410989458459250064, 7.51820061069283424252594651531, 8.457974035788773724892984699039, 10.33330499595731853882316047746, 11.48575195465140900296763242333, 11.86259456325753231354589669316, 14.15262829587852239657666215309, 14.70517269937317926959315843606