L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (2.08 − 0.803i)5-s + (−0.499 − 0.866i)6-s + (0.475 + 0.475i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (1.31 + 1.80i)10-s + 1.62·11-s + (0.707 − 0.707i)12-s + (0.323 − 1.20i)13-s + (−0.336 + 0.582i)14-s + (−1.80 + 1.31i)15-s + (0.500 − 0.866i)16-s + (2.66 − 0.715i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.933 − 0.359i)5-s + (−0.204 − 0.353i)6-s + (0.179 + 0.179i)7-s + (−0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.416 + 0.571i)10-s + 0.490·11-s + (0.204 − 0.204i)12-s + (0.0898 − 0.335i)13-s + (−0.0898 + 0.155i)14-s + (−0.466 + 0.339i)15-s + (0.125 − 0.216i)16-s + (0.647 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40459 + 0.770862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40459 + 0.770862i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (-2.08 + 0.803i)T \) |
| 19 | \( 1 + (-3.74 - 2.23i)T \) |
good | 7 | \( 1 + (-0.475 - 0.475i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + (-0.323 + 1.20i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.66 + 0.715i)T + (14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (1.29 + 0.346i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.85 - 4.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.96iT - 31T^{2} \) |
| 37 | \( 1 + (1.34 - 1.34i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.11 - 1.80i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.07 + 4.02i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (2.70 - 10.0i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.297 + 1.10i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.21 - 5.57i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.415 + 0.719i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.62 + 1.23i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.55 - 3.78i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.10 + 11.5i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.50 + 6.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.66 + 4.66i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.342 - 0.593i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.83 + 6.83i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.72803019073046544669424977230, −9.903365821498149322475921679448, −9.147681623715722820810747901699, −8.216776010043901515739085985206, −7.10489716133415519266778454696, −6.15071838999653393713366252112, −5.44896007379123923460892135691, −4.67866985716927995431646420863, −3.23826307758116137746167006079, −1.34102342813916424584412728811,
1.20893562339177576822504330201, 2.47006770830259023108734408286, 3.83769963099399578056992518242, 5.04023292889407329114477084000, 5.92037690502834630021867232233, 6.76963928721928241836209415876, 7.918979254025749031138801259573, 9.263564605484074267454612007549, 9.829545240379986852525568591248, 10.67632728417810999867252616619