L(s) = 1 | + (0.839 + 0.839i)3-s + (3.71 + 55.7i)5-s + (−99.3 + 99.3i)7-s − 241. i·9-s − 637. i·11-s + (−640. + 640. i)13-s + (−43.7 + 49.9i)15-s + (648. + 648. i)17-s − 2.50e3·19-s − 166.·21-s + (−2.80e3 − 2.80e3i)23-s + (−3.09e3 + 414. i)25-s + (406. − 406. i)27-s + 4.95e3i·29-s − 1.96e3i·31-s + ⋯ |
L(s) = 1 | + (0.0538 + 0.0538i)3-s + (0.0664 + 0.997i)5-s + (−0.766 + 0.766i)7-s − 0.994i·9-s − 1.58i·11-s + (−1.05 + 1.05i)13-s + (−0.0501 + 0.0573i)15-s + (0.544 + 0.544i)17-s − 1.59·19-s − 0.0825·21-s + (−1.10 − 1.10i)23-s + (−0.991 + 0.132i)25-s + (0.107 − 0.107i)27-s + 1.09i·29-s − 0.366i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0126866 - 0.262718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0126866 - 0.262718i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3.71 - 55.7i)T \) |
good | 3 | \( 1 + (-0.839 - 0.839i)T + 243iT^{2} \) |
| 7 | \( 1 + (99.3 - 99.3i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 637. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (640. - 640. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-648. - 648. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.80e3 + 2.80e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 4.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.96e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.89e3 + 1.89e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 5.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.06e4 - 1.06e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (8.30e3 - 8.30e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-7.43e3 + 7.43e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (4.15e3 - 4.15e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 1.22e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.60e4 + 3.60e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 6.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.28e4 - 6.28e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 2.44e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (8.96e4 + 8.96e4i)T + 8.58e9iT^{2} \) |
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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
−14.30582191580298017031461664274, −12.75189687183818733602584457372, −11.84141379520036653035254905496, −10.64792459505544757220350788199, −9.514509788330840945036052623760, −8.442424276902036986466932128064, −6.63382242044878872268587214514, −6.04367377506683502529429924457, −3.78309609853274643158912870320, −2.53306642534811098433529980698,
0.10223058955846150327380687089, 2.10215449842015770000186826855, 4.23103460245013066499069696131, 5.35949865974750092615512394716, 7.18318502173427707999993614278, 8.088289453661232120047277641295, 9.768888899926958429085861383196, 10.24342422048495668512224257600, 12.09179902208466950747015736442, 12.86440317463055806576115075942