L(s) = 1 | + (−0.141 + 1.99i)2-s + (−2.17 − 2.06i)3-s + (−3.95 − 0.565i)4-s + (4.43 − 4.04i)6-s + (−5.18 + 5.18i)7-s + (1.68 − 7.81i)8-s + (0.459 + 8.98i)9-s − 7.14·11-s + (7.44 + 9.41i)12-s + (7.93 − 7.93i)13-s + (−9.61 − 11.0i)14-s + (15.3 + 4.47i)16-s + (16.5 − 16.5i)17-s + (−17.9 − 0.357i)18-s + 12.1·19-s + ⋯ |
L(s) = 1 | + (−0.0708 + 0.997i)2-s + (−0.724 − 0.688i)3-s + (−0.989 − 0.141i)4-s + (0.738 − 0.674i)6-s + (−0.741 + 0.741i)7-s + (0.211 − 0.977i)8-s + (0.0510 + 0.998i)9-s − 0.649·11-s + (0.620 + 0.784i)12-s + (0.610 − 0.610i)13-s + (−0.686 − 0.791i)14-s + (0.960 + 0.279i)16-s + (0.975 − 0.975i)17-s + (−0.999 − 0.0198i)18-s + 0.639·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.901789 + 0.0521237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.901789 + 0.0521237i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.141 - 1.99i)T \) |
| 3 | \( 1 + (2.17 + 2.06i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (5.18 - 5.18i)T - 49iT^{2} \) |
| 11 | \( 1 + 7.14T + 121T^{2} \) |
| 13 | \( 1 + (-7.93 + 7.93i)T - 169iT^{2} \) |
| 17 | \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \) |
| 19 | \( 1 - 12.1T + 361T^{2} \) |
| 23 | \( 1 + (-11.0 + 11.0i)T - 529iT^{2} \) |
| 29 | \( 1 - 26.1T + 841T^{2} \) |
| 31 | \( 1 + 8.74iT - 961T^{2} \) |
| 37 | \( 1 + (-26.7 - 26.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 35.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.6 - 24.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (58.6 + 58.6i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-20.4 - 20.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 59.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 7.42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (35.8 - 35.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 46.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (10.6 - 10.6i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 68.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-76.6 + 76.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 41.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (81.7 + 81.7i)T + 9.40e3iT^{2} \) |
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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
−11.76409786837495076171059234826, −10.45809861667890282733119406562, −9.576030574923509315935534656026, −8.380705673129483619470298484096, −7.56076577522705304201604264151, −6.55424547396364212347968960831, −5.70888761200738839784258670694, −4.97532757596517535171695949524, −3.01793920932084124739508135670, −0.67845624081142449938648613249,
1.01819063518142826236924882975, 3.21097769886209245150874890289, 4.05583412453092864192944212818, 5.22773534970392431143540665393, 6.35383861468143884157615756582, 7.81884649570252338943416651964, 9.124784570893294356706547254957, 9.935523581118421323611054808877, 10.56201245906556761922399383200, 11.32095835946121785704096208283