| L(s) = 1 | + (−0.930 − 0.365i)2-s + (−1.72 + 0.170i)3-s + (0.733 + 0.680i)4-s + (−0.562 − 0.383i)5-s + (1.66 + 0.470i)6-s + (−1.02 + 2.43i)7-s + (−0.433 − 0.900i)8-s + (2.94 − 0.588i)9-s + (0.383 + 0.562i)10-s + (−0.601 − 3.99i)11-s + (−1.37 − 1.04i)12-s + (1.92 − 1.53i)13-s + (1.84 − 1.89i)14-s + (1.03 + 0.564i)15-s + (0.0747 + 0.997i)16-s + (−1.73 − 0.533i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 − 0.258i)2-s + (−0.995 + 0.0984i)3-s + (0.366 + 0.340i)4-s + (−0.251 − 0.171i)5-s + (0.680 + 0.192i)6-s + (−0.388 + 0.921i)7-s + (−0.153 − 0.318i)8-s + (0.980 − 0.196i)9-s + (0.121 + 0.177i)10-s + (−0.181 − 1.20i)11-s + (−0.398 − 0.302i)12-s + (0.534 − 0.426i)13-s + (0.493 − 0.505i)14-s + (0.267 + 0.145i)15-s + (0.0186 + 0.249i)16-s + (−0.419 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338364 - 0.355096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338364 - 0.355096i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.72 - 0.170i)T \) |
| 7 | \( 1 + (1.02 - 2.43i)T \) |
| good | 5 | \( 1 + (0.562 + 0.383i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.601 + 3.99i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-1.92 + 1.53i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.73 + 0.533i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.85 + 1.07i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.52 + 8.18i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-5.96 + 1.36i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (4.37 + 2.52i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.91 + 6.41i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-6.53 + 3.14i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.51 - 0.731i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.27 - 10.8i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (7.86 - 8.47i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.00789 + 0.00538i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-2.82 - 3.04i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.82 + 6.62i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.04 - 1.15i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.71 - 1.84i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 2.65i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.403 + 0.505i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (10.5 + 1.59i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 9.68iT - 97T^{2} \) |
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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
−11.29859152562754003994500346693, −10.82174916609698920116837522442, −9.709388179310784350046583557906, −8.756768699794672011623185273145, −7.86735502423499690534422642682, −6.36208140512089357706814704072, −5.82535436435061000069017418635, −4.30851553597905376709892701296, −2.71347619616674022555857798952, −0.53962522630119887493426995883,
1.49372033853237993735212106813, 3.82999263528893860145471230006, 5.08787460392524734804049753567, 6.39253763962127277529581034697, 7.13902411980869708454314602012, 7.87275799200593829670609509823, 9.541976023609433891001247579765, 10.05192152285243459551064075724, 11.08017638094524690174556180206, 11.70199908513029151149343766307
