L(s) = 1 | + (2.74 + 1.58i)3-s − 1.53i·7-s + (3.50 + 6.07i)9-s − 3.79·11-s + (5.80 − 3.34i)13-s + (5.21 + 3.00i)17-s + (3.93 + 1.87i)19-s + (2.42 − 4.20i)21-s + (−3.79 + 2.19i)23-s + 12.7i·27-s + (−0.549 − 0.951i)29-s − 1.05·31-s + (−10.4 − 6.00i)33-s − 10.7i·37-s + 21.2·39-s + ⋯ |
L(s) = 1 | + (1.58 + 0.913i)3-s − 0.579i·7-s + (1.16 + 2.02i)9-s − 1.14·11-s + (1.60 − 0.928i)13-s + (1.26 + 0.729i)17-s + (0.902 + 0.430i)19-s + (0.529 − 0.917i)21-s + (−0.791 + 0.457i)23-s + 2.44i·27-s + (−0.102 − 0.176i)29-s − 0.189·31-s + (−1.81 − 1.04i)33-s − 1.77i·37-s + 3.39·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.264659056\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.264659056\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.93 - 1.87i)T \) |
good | 3 | \( 1 + (-2.74 - 1.58i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 1.53iT - 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + (-5.80 + 3.34i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.21 - 3.00i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.79 - 2.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.549 + 0.951i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 - 2.77i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.73 - 5.61i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.36 - 0.788i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 2.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0331 + 0.0574i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.94 - 2.85i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.23 + 5.60i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 6.98i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.996 - 1.72i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.68 + 4.43i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.422052391660893406638449927850, −8.340857857648889121134200060240, −7.971153138224708653351433222070, −7.50536050311709730565317720291, −5.91060290600904364191707961095, −5.22523519357008488293452040383, −3.92348699494138031987568888339, −3.57535708953068813737694857810, −2.73287822909256848874783519886, −1.40585049020022463081376557475,
1.16765070659586012016622681824, 2.18406479684249371499446573633, 3.05772450063551588369520005288, 3.70419656303561619406083435916, 5.07371169458476901151469513125, 6.11413203188529973187833783085, 6.94208064702522864738224360030, 7.74814489867646480000028065876, 8.283485506290429033222173836544, 8.915863146935191669504145255773