L(s) = 1 | + (−0.433 − 0.596i)2-s + (2.67 − 0.868i)3-s + (0.449 − 1.38i)4-s + (−1.67 − 1.21i)6-s + (−0.980 − 0.318i)7-s + (−2.42 + 0.787i)8-s + (3.96 − 2.88i)9-s + (1.93 + 2.69i)11-s − 4.09i·12-s + (−2.02 − 2.79i)13-s + (0.235 + 0.723i)14-s + (−0.834 − 0.606i)16-s + (−1.40 + 1.94i)17-s + (−3.44 − 1.11i)18-s + (2.36 + 7.29i)19-s + ⋯ |
L(s) = 1 | + (−0.306 − 0.421i)2-s + (1.54 − 0.501i)3-s + (0.224 − 0.692i)4-s + (−0.684 − 0.497i)6-s + (−0.370 − 0.120i)7-s + (−0.857 + 0.278i)8-s + (1.32 − 0.961i)9-s + (0.583 + 0.811i)11-s − 1.18i·12-s + (−0.562 − 0.773i)13-s + (0.0628 + 0.193i)14-s + (−0.208 − 0.151i)16-s + (−0.341 + 0.470i)17-s + (−0.811 − 0.263i)18-s + (0.543 + 1.67i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29670 - 1.14859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29670 - 1.14859i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (-1.93 - 2.69i)T \) |
good | 2 | \( 1 + (0.433 + 0.596i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-2.67 + 0.868i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (0.980 + 0.318i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2.02 + 2.79i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.40 - 1.94i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.36 - 7.29i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 2.45iT - 23T^{2} \) |
| 29 | \( 1 + (-1.83 + 5.66i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.98 + 2.16i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (5.66 + 1.84i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.64iT - 43T^{2} \) |
| 47 | \( 1 + (-5.55 + 1.80i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.96 - 9.58i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.910 - 2.80i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 6.14iT - 67T^{2} \) |
| 71 | \( 1 + (1.63 + 1.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.785 + 0.255i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (9.77 - 7.09i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.946 + 1.30i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 + (-1.43 - 1.97i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.86249545037669069862247655377, −10.25937904441000380666537824297, −9.834933803408437989694293011489, −8.917462203067238784261535733052, −7.909689054319357506798823551724, −6.99788016541028212328277950716, −5.75772080995362184082585502275, −3.91490782899396041680177567326, −2.65994058571815533326523097717, −1.56910256551485570150278275458,
2.61644220540105824637604222727, 3.37174242496646949265376430468, 4.63254426480577726181498267889, 6.61976737855160213165409667892, 7.33585267068394219910925364390, 8.586853255311971377493549588860, 8.949922136430979784858121805578, 9.701539502991045562466168892760, 11.15141656003221163133711948241, 12.15571083792512274130018538454