L(s) = 1 | + (−1.08 + 1.67i)2-s − 1.73·3-s + (−1.64 − 3.64i)4-s + (1.87 − 2.90i)6-s + 0.596·7-s + (7.91 + 1.19i)8-s + 2.99·9-s − 9.27i·11-s + (2.84 + 6.31i)12-s + 23.5i·13-s + (−0.647 + 1.00i)14-s + (−10.5 + 11.9i)16-s + 3.97i·17-s + (−3.25 + 5.03i)18-s − 7.04i·19-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.839i)2-s − 0.577·3-s + (−0.410 − 0.911i)4-s + (0.313 − 0.484i)6-s + 0.0852·7-s + (0.988 + 0.149i)8-s + 0.333·9-s − 0.843i·11-s + (0.237 + 0.526i)12-s + 1.80i·13-s + (−0.0462 + 0.0715i)14-s + (−0.662 + 0.749i)16-s + 0.233i·17-s + (−0.180 + 0.279i)18-s − 0.370i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00820418 + 0.408814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00820418 + 0.408814i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.08 - 1.67i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.596T + 49T^{2} \) |
| 11 | \( 1 + 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 23.5iT - 169T^{2} \) |
| 17 | \( 1 - 3.97iT - 289T^{2} \) |
| 19 | \( 1 + 7.04iT - 361T^{2} \) |
| 23 | \( 1 + 32.0T + 529T^{2} \) |
| 29 | \( 1 + 35.6T + 841T^{2} \) |
| 31 | \( 1 - 59.2iT - 961T^{2} \) |
| 37 | \( 1 + 5.38iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 36.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 74.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.55iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 36.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.73T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 59.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 83.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 65.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 129.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 130.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.1iT - 9.40e3T^{2} \) |
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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
−11.67512075021026758828939775357, −11.04388093208420473558831289415, −9.954001622468304805194404610950, −9.072808299959063640418077085723, −8.178316059334748283469934256438, −6.96385380375212400001050727214, −6.27634177813643042935962645526, −5.20765275237015429492367852930, −4.05124651232778023108469142368, −1.62688537757731947559648723965,
0.25770002253266983709640297545, 1.98704329632274732566181784947, 3.52251686373217960187460707348, 4.77912856142762448735957758025, 5.99592163964137142706714917112, 7.56063884858972284678670145335, 8.081461610290299811878943795817, 9.617486739119830394519076025482, 10.07899986613063208570106987668, 11.06216065509057297865828860111