| L(s) = 1 | + (0.906 + 0.422i)2-s + (−1.16 + 1.28i)3-s + (0.642 + 0.766i)4-s + (0.318 − 2.21i)5-s + (−1.59 + 0.674i)6-s + (−2.88 − 0.771i)7-s + (0.258 + 0.965i)8-s + (−0.303 − 2.98i)9-s + (1.22 − 1.87i)10-s + (−4.54 − 2.62i)11-s + (−1.73 − 0.0633i)12-s + (−3.61 + 5.15i)13-s + (−2.28 − 1.91i)14-s + (2.47 + 2.97i)15-s + (−0.173 + 0.984i)16-s + (−1.82 − 0.850i)17-s + ⋯ |
| L(s) = 1 | + (0.640 + 0.298i)2-s + (−0.670 + 0.742i)3-s + (0.321 + 0.383i)4-s + (0.142 − 0.989i)5-s + (−0.651 + 0.275i)6-s + (−1.08 − 0.291i)7-s + (0.0915 + 0.341i)8-s + (−0.101 − 0.994i)9-s + (0.387 − 0.591i)10-s + (−1.36 − 0.790i)11-s + (−0.499 − 0.0182i)12-s + (−1.00 + 1.43i)13-s + (−0.610 − 0.512i)14-s + (0.638 + 0.769i)15-s + (−0.0434 + 0.246i)16-s + (−0.442 − 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0880997 - 0.193159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0880997 - 0.193159i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.906 - 0.422i)T \) |
| 3 | \( 1 + (1.16 - 1.28i)T \) |
| 5 | \( 1 + (-0.318 + 2.21i)T \) |
| 19 | \( 1 + (3.51 + 2.57i)T \) |
| good | 7 | \( 1 + (2.88 + 0.771i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.54 + 2.62i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.61 - 5.15i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.82 + 0.850i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (0.294 - 3.36i)T + (-22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-4.14 - 1.51i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.72 + 2.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.88 + 3.88i)T - 37iT^{2} \) |
| 41 | \( 1 + (-8.88 - 1.56i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.503 + 5.74i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.97 - 8.51i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.925 + 10.5i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (4.19 - 1.52i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.74 - 7.33i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (8.67 - 4.04i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (1.24 - 1.48i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-10.8 + 7.61i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (6.95 + 1.22i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.36 + 0.902i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.58 + 8.97i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.58 - 11.9i)T + (-62.3 - 74.3i)T^{2} \) |
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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
−10.48075163120759085378976868452, −9.480559975006797148904957097979, −8.924891595535466390274693835490, −7.54837105440018591211998955503, −6.48095175326117795777845325185, −5.68721607336696176445080781217, −4.76011475955973834079234283114, −4.07421833722111500232135450372, −2.66017956723498628394060331945, −0.094890264404554241862985398723,
2.38960045555534951810755188148, 2.88726189740007898114632181594, 4.64311862291504466662350740165, 5.72447137524746314856648187998, 6.35393661756167552827835873199, 7.27483953111027672922479155112, 8.016784594607908684159487011189, 9.834436657957013458386596944563, 10.42656341628824643535262122416, 10.89966974651113335558339415028
