L(s) = 1 | + (−0.866 + 0.5i)2-s + (−1.38 − 1.03i)3-s + (0.499 − 0.866i)4-s + (−1.75 + 3.03i)5-s + (1.72 + 0.201i)6-s + (1.37 + 2.26i)7-s + 0.999i·8-s + (0.858 + 2.87i)9-s − 3.50i·10-s − 3.53i·11-s + (−1.59 + 0.685i)12-s + (−3.59 − 0.316i)13-s + (−2.32 − 1.27i)14-s + (5.57 − 2.40i)15-s + (−0.5 − 0.866i)16-s + (1.97 − 3.42i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.801 − 0.597i)3-s + (0.249 − 0.433i)4-s + (−0.784 + 1.35i)5-s + (0.702 + 0.0823i)6-s + (0.519 + 0.854i)7-s + 0.353i·8-s + (0.286 + 0.958i)9-s − 1.10i·10-s − 1.06i·11-s + (−0.459 + 0.197i)12-s + (−0.996 − 0.0876i)13-s + (−0.620 − 0.339i)14-s + (1.44 − 0.620i)15-s + (−0.125 − 0.216i)16-s + (0.479 − 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0217084 - 0.0906817i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0217084 - 0.0906817i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (1.38 + 1.03i)T \) |
| 7 | \( 1 + (-1.37 - 2.26i)T \) |
| 13 | \( 1 + (3.59 + 0.316i)T \) |
good | 5 | \( 1 + (1.75 - 3.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 17 | \( 1 + (-1.97 + 3.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 6.65iT - 19T^{2} \) |
| 23 | \( 1 + (-1.11 + 0.642i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.71 + 3.29i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (8.14 - 4.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.28 + 9.15i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.84 - 4.92i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.79 + 4.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0582 - 0.100i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.70 - 3.87i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.84 + 4.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + (8.18 - 4.72i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.73 + 2.15i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.63 - 4.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.03T + 83T^{2} \) |
| 89 | \( 1 + (0.177 + 0.307i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.507 - 0.292i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.19571612410163450900976962798, −10.69791181603376762153047907433, −9.630603404507824102346952044167, −8.313044753775106097650462097456, −7.60618801076334934745791442569, −7.00146345037808265584585240319, −5.91836792610622789186194867021, −5.24131342388629173026676562606, −3.38927437431671370276743646784, −2.04539653380437196431582213952,
0.07409787292345103485693082883, 1.49622963737799283384043262418, 3.74682306024170494410356172782, 4.62900066641577523480328455754, 5.18619016986600266078801878934, 6.99417370957372973046922628596, 7.59990593770230695980264421697, 8.731633509232160061564882060998, 9.502317794082808576386592571877, 10.27412939659256279639386145171