L(s) = 1 | + (1.12 − 2.20i)2-s + (0.382 − 1.68i)3-s + (−2.43 − 3.35i)4-s + (−3.30 − 2.74i)6-s + (0.134 − 0.847i)7-s + (−5.26 + 0.833i)8-s + (−2.70 − 1.29i)9-s + (−3.29 + 0.418i)11-s + (−6.60 + 2.83i)12-s + (−2.14 + 4.20i)13-s + (−1.72 − 1.25i)14-s + (−1.52 + 4.67i)16-s + (4.82 − 2.45i)17-s + (−5.90 + 4.52i)18-s + (2.92 − 4.03i)19-s + ⋯ |
L(s) = 1 | + (0.796 − 1.56i)2-s + (0.220 − 0.975i)3-s + (−1.21 − 1.67i)4-s + (−1.34 − 1.12i)6-s + (0.0507 − 0.320i)7-s + (−1.86 + 0.294i)8-s + (−0.902 − 0.430i)9-s + (−0.992 + 0.126i)11-s + (−1.90 + 0.818i)12-s + (−0.594 + 1.16i)13-s + (−0.459 − 0.334i)14-s + (−0.380 + 1.16i)16-s + (1.17 − 0.596i)17-s + (−1.39 + 1.06i)18-s + (0.672 − 0.924i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975758 + 1.57318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975758 + 1.57318i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.382 + 1.68i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (3.29 - 0.418i)T \) |
good | 2 | \( 1 + (-1.12 + 2.20i)T + (-1.17 - 1.61i)T^{2} \) |
| 7 | \( 1 + (-0.134 + 0.847i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (2.14 - 4.20i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.82 + 2.45i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 + 4.03i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.93 - 1.93i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.756 + 0.549i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.95 + 6.02i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.46 + 9.22i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.90 + 4.00i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (2.32 + 2.32i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.85 - 11.6i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (1.46 + 0.748i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (2.46 - 1.79i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.12 + 9.60i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.03 + 6.03i)T - 67iT^{2} \) |
| 71 | \( 1 + (1.61 + 0.524i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 1.77i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (11.4 - 3.72i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.17 - 12.1i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + (8.91 + 4.54i)T + (57.0 + 78.4i)T^{2} \) |
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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
−9.668968422768521264434500671969, −9.300578024074789370506977264478, −7.79365415388696315240248058230, −7.19019267300763273283585138147, −5.79013714329113103478032815845, −5.00214001752591424411228615749, −3.86832561843380277665970965896, −2.76742306377803243947985942663, −2.04902946120756832406623939974, −0.67785647554482460750647726833,
2.93690789231097613521155446466, 3.71693306723521821277592548300, 5.00051744082971385620957264851, 5.37622835595962945891228808123, 6.12960329544842800434421876578, 7.53033835592018852935208806229, 8.072189771454977921207736671997, 8.677400802861096248728923972655, 10.05978812699420969533685646611, 10.35484033003955640897216267215