L(s) = 1 | + (−0.461 − 1.72i)2-s + (0.965 + 0.258i)3-s + (−1.01 + 0.587i)4-s + (−1.92 − 1.14i)5-s − 1.78i·6-s + (2.04 − 1.68i)7-s + (−1.03 − 1.03i)8-s + (0.866 + 0.499i)9-s + (−1.07 + 3.83i)10-s + (1.46 + 2.54i)11-s + (−1.13 + 0.304i)12-s + (0.187 − 0.187i)13-s + (−3.83 − 2.74i)14-s + (−1.56 − 1.60i)15-s + (−2.48 + 4.30i)16-s + (−0.868 + 3.24i)17-s + ⋯ |
L(s) = 1 | + (−0.326 − 1.21i)2-s + (0.557 + 0.149i)3-s + (−0.508 + 0.293i)4-s + (−0.859 − 0.510i)5-s − 0.727i·6-s + (0.772 − 0.635i)7-s + (−0.367 − 0.367i)8-s + (0.288 + 0.166i)9-s + (−0.341 + 1.21i)10-s + (0.442 + 0.766i)11-s + (−0.327 + 0.0877i)12-s + (0.0521 − 0.0521i)13-s + (−1.02 − 0.732i)14-s + (−0.403 − 0.413i)15-s + (−0.621 + 1.07i)16-s + (−0.210 + 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.598465 - 0.783553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.598465 - 0.783553i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (1.92 + 1.14i)T \) |
| 7 | \( 1 + (-2.04 + 1.68i)T \) |
good | 2 | \( 1 + (0.461 + 1.72i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-1.46 - 2.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.187 + 0.187i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.868 - 3.24i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.81 - 3.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-9.07 + 2.43i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 0.815iT - 29T^{2} \) |
| 31 | \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.69 + 6.31i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.45iT - 41T^{2} \) |
| 43 | \( 1 + (3.59 + 3.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.24 - 2.47i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.30 - 4.87i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.07 - 4.66i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (15.3 + 4.10i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 + (-1.89 - 0.508i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.44 - 2.56i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.09 - 6.09i)T - 83iT^{2} \) |
| 89 | \( 1 + (-4.87 + 8.43i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.93 - 5.93i)T + 97iT^{2} \) |
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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
−13.02010635872192766550725879999, −12.35837842030919188486971741064, −11.19912148273915436003588755993, −10.51645332923214982842893274995, −9.223898425974067538482883752796, −8.319835926699969714041383726567, −7.04397429959187457822540332212, −4.60587178749344161778496142921, −3.55620999852765921062475031024, −1.57878640858291730237383684010,
2.99357333005241772563751415647, 4.97661347870897894904430076015, 6.56213330025881968341220603487, 7.46074695033761143225537562993, 8.456062408216140944108717386270, 9.121670431107793877789139364780, 11.16534286562938909705197004532, 11.75585329067803323900688359558, 13.40217551265866739082646390144, 14.61323719007041269296380912738