L(s) = 1 | + (−0.459 − 1.48i)2-s + (−0.910 + 1.47i)3-s + (−0.352 + 0.240i)4-s + (−3.51 − 0.529i)5-s + (2.61 + 0.679i)6-s + (−2.23 + 1.41i)7-s + (−1.91 − 1.52i)8-s + (−1.34 − 2.68i)9-s + (0.825 + 5.47i)10-s + (0.609 − 0.657i)11-s + (−0.0329 − 0.737i)12-s + (−3.35 − 0.766i)13-s + (3.13 + 2.67i)14-s + (3.98 − 4.69i)15-s + (−1.70 + 4.34i)16-s + (−0.307 + 4.10i)17-s + ⋯ |
L(s) = 1 | + (−0.324 − 1.05i)2-s + (−0.525 + 0.850i)3-s + (−0.176 + 0.120i)4-s + (−1.57 − 0.236i)5-s + (1.06 + 0.277i)6-s + (−0.844 + 0.535i)7-s + (−0.677 − 0.540i)8-s + (−0.446 − 0.894i)9-s + (0.260 + 1.73i)10-s + (0.183 − 0.198i)11-s + (−0.00952 − 0.212i)12-s + (−0.931 − 0.212i)13-s + (0.838 + 0.714i)14-s + (1.02 − 1.21i)15-s + (−0.426 + 1.08i)16-s + (−0.0745 + 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0246972 + 0.0942370i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0246972 + 0.0942370i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.910 - 1.47i)T \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
good | 2 | \( 1 + (0.459 + 1.48i)T + (-1.65 + 1.12i)T^{2} \) |
| 5 | \( 1 + (3.51 + 0.529i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-0.609 + 0.657i)T + (-0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.35 + 0.766i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.307 - 4.10i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-5.05 + 2.92i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.000984 - 7.37e-5i)T + (22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (0.494 + 1.02i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (8.65 + 4.99i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.03 + 3.43i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (2.38 - 2.99i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (1.80 + 2.26i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-5.59 + 1.72i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (-0.765 - 1.12i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (9.22 - 1.39i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (2.68 - 3.94i)T + (-22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (0.431 - 0.746i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.29 - 6.84i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.66 - 11.8i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-3.99 - 6.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.48 + 15.2i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-6.35 + 5.89i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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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
−12.06787731854679373374545101521, −11.51807621961259434102853739733, −10.60497015920683611356153013642, −9.582182186870375885339224398375, −8.772531017394860044111034205430, −7.15272711656367823362867007204, −5.64678129815978839760056956484, −4.04150900623131613490215767389, −3.11950388580932570176868203186, −0.10749343728441288590870977975,
3.24483476481012385139153946790, 5.12493842372472132695620702694, 6.66074930831244724594342033266, 7.33403116962247458701711422152, 7.76113184934492706501819097225, 9.214534881499725088973771447814, 10.81161731859490069707899041655, 12.01154909419223915352701448420, 12.21487701173986796792919161048, 13.80597593335002768895032531635