| L(s) = 1 | + (−0.930 − 0.365i)2-s + (−0.560 + 1.63i)3-s + (0.733 + 0.680i)4-s + (−1.71 − 1.16i)5-s + (1.12 − 1.32i)6-s + (1.49 − 2.18i)7-s + (−0.433 − 0.900i)8-s + (−2.37 − 1.83i)9-s + (1.16 + 1.71i)10-s + (−0.152 − 1.01i)11-s + (−1.52 + 0.820i)12-s + (1.05 − 0.843i)13-s + (−2.19 + 1.48i)14-s + (2.86 − 2.14i)15-s + (0.0747 + 0.997i)16-s + (4.47 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 − 0.258i)2-s + (−0.323 + 0.946i)3-s + (0.366 + 0.340i)4-s + (−0.764 − 0.521i)5-s + (0.457 − 0.539i)6-s + (0.566 − 0.824i)7-s + (−0.153 − 0.318i)8-s + (−0.790 − 0.612i)9-s + (0.368 + 0.540i)10-s + (−0.0460 − 0.305i)11-s + (−0.440 + 0.236i)12-s + (0.293 − 0.233i)13-s + (−0.585 + 0.396i)14-s + (0.740 − 0.555i)15-s + (0.0186 + 0.249i)16-s + (1.08 + 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.661554 - 0.311981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.661554 - 0.311981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 + (0.560 - 1.63i)T \) |
| 7 | \( 1 + (-1.49 + 2.18i)T \) |
| good | 5 | \( 1 + (1.71 + 1.16i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.152 + 1.01i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.05 + 0.843i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.47 - 1.38i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-4.60 + 2.66i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.42 + 4.61i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.16 + 0.949i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 0.762i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.35 - 4.97i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.259 - 0.125i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (9.14 + 4.40i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-3.37 + 8.61i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-6.36 + 6.86i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (4.07 - 2.77i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-8.44 - 9.09i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (4.21 - 7.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.45 + 0.789i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (8.01 - 3.14i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (3.94 + 6.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.57 - 9.50i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-11.2 - 1.69i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 0.848iT - 97T^{2} \) |
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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
−11.76106391972243590312823562464, −10.41469318139655855371563841641, −10.13525187672236494925368811950, −8.656636544868719208104429903824, −8.193384197481369206279995625131, −6.94981881395774715230699992657, −5.40257883319718753220854820790, −4.31131561009204097638711335920, −3.30185770695254178604685528670, −0.77611652940082853352898487876,
1.54475957161645192039866218555, 3.14094214025441363524976512406, 5.20569493976253895065874696731, 6.12965195973622692530662265403, 7.42821687152731247546719853425, 7.76015559056468022923512674154, 8.823881440722065996155649384866, 10.02852399317638160512249765493, 11.25366634019665182802969379013, 11.76781452832290773811051536219
