L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (−2.62 + 0.358i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (1.5 + 2.59i)11-s + (−0.499 + 0.866i)12-s − 1.41·13-s + (1 − 2.44i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + (−0.990 + 0.135i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.452 + 0.783i)11-s + (−0.144 + 0.249i)12-s − 0.392·13-s + (0.267 − 0.654i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.450638 + 0.550802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.450638 + 0.550802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.62 - 0.358i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.292 + 0.507i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 31 | \( 1 + (-3.32 - 5.76i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.53 - 4.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 + (6.65 - 11.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.91 - 6.77i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.62 - 8.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.82 - 13.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + (-2.82 - 4.89i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.20 - 2.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + (-5.36 + 9.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.41T + 97T^{2} \) |
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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
−9.948939470589015210106956596588, −9.359164813726855587170163971127, −8.696618274312697202181003820211, −7.43788129136831696033791254681, −6.99734403473974909215378474809, −6.09097071538205884548705388954, −5.29317422911654618217618339367, −4.24118741443182359269470920858, −2.71901848258866962937637022354, −1.26940145688420037913210240131,
0.41796732134421822642017607744, 2.29490781854906930480249292512, 3.42107707822748857855838296284, 4.09544115576368824465427532432, 5.50581256005384880812755183741, 6.32120569169977036863396451410, 7.21884720926094081766596283022, 8.357624177720393559858776672804, 9.313487723417173909486896397162, 9.754498837625337846517425415696