| L(s) = 1 | + (−1.03 − 0.866i)2-s + (2.70 − 0.984i)3-s + (−0.0320 − 0.181i)4-s + (0.152 − 0.866i)5-s + (−3.64 − 1.32i)6-s + (−0.173 − 0.300i)7-s + (−1.47 + 2.54i)8-s + (4.05 − 3.40i)9-s + (−0.907 + 0.761i)10-s + (1.11 − 1.92i)11-s + (−0.266 − 0.460i)12-s + (−2.41 − 0.880i)13-s + (−0.0812 + 0.460i)14-s + (−0.439 − 2.49i)15-s + (3.37 − 1.22i)16-s + (0.358 + 0.300i)17-s + ⋯ |
| L(s) = 1 | + (−0.729 − 0.612i)2-s + (1.56 − 0.568i)3-s + (−0.0160 − 0.0909i)4-s + (0.0682 − 0.387i)5-s + (−1.48 − 0.541i)6-s + (−0.0656 − 0.113i)7-s + (−0.520 + 0.901i)8-s + (1.35 − 1.13i)9-s + (−0.287 + 0.240i)10-s + (0.335 − 0.581i)11-s + (−0.0768 − 0.133i)12-s + (−0.670 − 0.244i)13-s + (−0.0217 + 0.123i)14-s + (−0.113 − 0.643i)15-s + (0.844 − 0.307i)16-s + (0.0869 + 0.0729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.724883 - 1.25458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.724883 - 1.25458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (1.03 + 0.866i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-2.70 + 0.984i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.152 + 0.866i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.300i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.11 + 1.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.41 + 0.880i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.358 - 0.300i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.467 + 2.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.26 - 4.42i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.55 - 6.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 + (2.32 - 0.846i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.677 + 3.84i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.58 - 4.68i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.492 + 2.79i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.83 + 4.05i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.58 - 8.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.87 + 4.93i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.61 - 9.16i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.30 - 0.475i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.1 + 4.05i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.41 - 12.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.67 - 3.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.24 - 6.07i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.91920872118916561518555127827, −9.956136245697113975811363610850, −9.126732541631269530070273387258, −8.609040301169370365445245403921, −7.79187354227824195326323633643, −6.63314112065316689564121300200, −5.11049851311927378754126506968, −3.43577086366452712680440356581, −2.40387679045036001105541194947, −1.17431357306883297853017935726,
2.35973267346826351318301140569, 3.47889058210339725863452411004, 4.50684818966749780723700394509, 6.37847706781013763753818904233, 7.50862432311588829812069535135, 7.945953165096908002815831400395, 9.098028495016172726414939438368, 9.504110830144660431550350155227, 10.23527981471730525503198823811, 11.71402403797672583285053244776
