L(s) = 1 | + (−7.79 − 13.5i)2-s + (13.5 + 7.79i)3-s + (−89.6 + 155. i)4-s + (−22.3 + 12.9i)5-s − 243. i·6-s + 1.79e3·8-s + (121.5 + 210. i)9-s + (348. + 201. i)10-s + (−311. + 540. i)11-s + (−2.41e3 + 1.39e3i)12-s − 3.25e3i·13-s − 402.·15-s + (−8.27e3 − 1.43e4i)16-s + (−275. − 158. i)17-s + (1.89e3 − 3.28e3i)18-s + (5.19e3 − 3.00e3i)19-s + ⋯ |
L(s) = 1 | + (−0.974 − 1.68i)2-s + (0.5 + 0.288i)3-s + (−1.40 + 2.42i)4-s + (−0.178 + 0.103i)5-s − 1.12i·6-s + 3.50·8-s + (0.166 + 0.288i)9-s + (0.348 + 0.201i)10-s + (−0.234 + 0.405i)11-s + (−1.40 + 0.808i)12-s − 1.48i·13-s − 0.119·15-s + (−2.02 − 3.49i)16-s + (−0.0560 − 0.0323i)17-s + (0.324 − 0.562i)18-s + (0.757 − 0.437i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4000430745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4000430745\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-13.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (7.79 + 13.5i)T + (-32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (22.3 - 12.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (311. - 540. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.25e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (275. + 158. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.19e3 + 3.00e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-21.0 - 36.3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.42e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.75e4 - 1.01e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-8.68e3 - 1.50e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.78e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (5.70e4 - 3.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (4.96e4 - 8.59e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (8.71e4 + 5.03e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.01e5 + 1.16e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.05e4 - 3.55e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 4.00e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (4.79e5 + 2.76e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-5.24e3 - 9.07e3i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 7.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.64e5 + 9.52e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.07e6iT - 8.32e11T^{2} \) |
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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
−11.80521722124247557270292105049, −10.93090057229470683313371887295, −10.03411598652901465738736392472, −9.364192378779078918518841493980, −8.193533325857616622544128247961, −7.52502113485251122697024755417, −4.91528117441195955762495547011, −3.50981227811407635977942529557, −2.72363926594664791243632866602, −1.28737822696663323301445944971,
0.17152523095953566220513957393, 1.67303280729078557028447042567, 4.21656101287895391564004674142, 5.62632220432911732903051011526, 6.70047255968686075179198247010, 7.57266026609390598637610843794, 8.481394259109586747305552602146, 9.267880734974254689295351583442, 10.18162279373501749431710497120, 11.63314655736396598827520185649