| 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 | \( 1 + (5.58 + 11.9i)T + (-62.3 + 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89966974651113335558339415028, −10.42656341628824643535262122416, −9.834436657957013458386596944563, −8.016784594607908684159487011189, −7.27483953111027672922479155112, −6.35393661756167552827835873199, −5.72447137524746314856648187998, −4.64311862291504466662350740165, −2.88726189740007898114632181594, −2.38960045555534951810755188148,
0.094890264404554241862985398723, 2.66017956723498628394060331945, 4.07421833722111500232135450372, 4.76011475955973834079234283114, 5.68721607336696176445080781217, 6.48095175326117795777845325185, 7.54837105440018591211998955503, 8.924891595535466390274693835490, 9.480559975006797148904957097979, 10.48075163120759085378976868452
