L(s) = 1 | + (−1.10 − 0.880i)2-s + (0.424 + 1.67i)3-s + (0.450 + 1.94i)4-s + (−0.5 − 0.866i)5-s + (1.00 − 2.23i)6-s + (−3.88 − 2.24i)7-s + (1.21 − 2.55i)8-s + (−2.63 + 1.42i)9-s + (−0.208 + 1.39i)10-s + (1.05 + 0.606i)11-s + (−3.08 + 1.58i)12-s + (−1.71 + 0.987i)13-s + (2.32 + 5.90i)14-s + (1.24 − 1.20i)15-s + (−3.59 + 1.75i)16-s − 5.28i·17-s + ⋯ |
L(s) = 1 | + (−0.782 − 0.622i)2-s + (0.245 + 0.969i)3-s + (0.225 + 0.974i)4-s + (−0.223 − 0.387i)5-s + (0.411 − 0.911i)6-s + (−1.46 − 0.847i)7-s + (0.430 − 0.902i)8-s + (−0.879 + 0.475i)9-s + (−0.0660 + 0.442i)10-s + (0.316 + 0.182i)11-s + (−0.889 + 0.457i)12-s + (−0.474 + 0.273i)13-s + (0.621 + 1.57i)14-s + (0.320 − 0.311i)15-s + (−0.898 + 0.438i)16-s − 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0389965 - 0.192597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0389965 - 0.192597i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.10 + 0.880i)T \) |
| 3 | \( 1 + (-0.424 - 1.67i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (3.88 + 2.24i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.05 - 0.606i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.71 - 0.987i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.28iT - 17T^{2} \) |
| 19 | \( 1 + 4.86T + 19T^{2} \) |
| 23 | \( 1 + (1.40 + 2.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.20 + 7.28i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.04 - 3.49i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.71iT - 37T^{2} \) |
| 41 | \( 1 + (5.30 - 3.06i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.17 + 2.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.97 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.739T + 53T^{2} \) |
| 59 | \( 1 + (-1.70 + 0.986i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.81 - 2.77i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.58 - 2.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + (-11.2 - 6.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.00 + 2.89i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2.71iT - 89T^{2} \) |
| 97 | \( 1 + (-8.61 + 14.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.77015997552359458211011793148, −9.939079256381928920552659771110, −9.473301124213333925503936718640, −8.620038873138905508246429796212, −7.44427144333397897055719357091, −6.44622751565950791254236531451, −4.57301716610455687928270401780, −3.75799883715054384111397932974, −2.63802169179543586588466179407, −0.15528127615440519813777470247,
2.03155042987103453081508940355, 3.36464290949910145338776249299, 5.60815333621776115203225710470, 6.43511937964532221059635952818, 6.95724363618708101130027422704, 8.203524873785643109766416611049, 8.826398728831890429798462513230, 9.793962861771291009096642204593, 10.74476011092075812796264275329, 11.93688066696054097914386685006