| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−202. + 116. i)5-s + (95.5 − 165. i)7-s − 181. i·8-s + 1.32e3·10-s + (673. + 388. i)11-s + (45.5 + 78.9i)13-s + (−936. + 540. i)14-s + (−512. + 886. i)16-s − 7.04e3i·17-s + 2.73e3·19-s + (−6.47e3 − 3.73e3i)20-s + (−2.19e3 − 3.80e3i)22-s + (1.72e4 − 9.94e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.61 + 0.934i)5-s + (0.278 − 0.482i)7-s − 0.353i·8-s + 1.32·10-s + (0.505 + 0.291i)11-s + (0.0207 + 0.0359i)13-s + (−0.341 + 0.197i)14-s + (−0.125 + 0.216i)16-s − 1.43i·17-s + 0.398·19-s + (−0.809 − 0.467i)20-s + (−0.206 − 0.357i)22-s + (1.41 − 0.817i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.684621 - 0.448019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.684621 - 0.448019i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (202. - 116. i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-95.5 + 165. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-673. - 388. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-45.5 - 78.9i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 7.04e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.73e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.72e4 + 9.94e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.71e4 + 1.56e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-6.17e3 - 1.06e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.79e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.74e4 + 2.16e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.92e4 + 3.33e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.43e5 - 8.30e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 5.47e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.41e4 + 8.14e3i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.94e4 + 5.09e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.47e5 + 2.56e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.57e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 8.02e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.88e5 + 3.26e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (7.33e5 + 4.23e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.12e3iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.75e5 + 1.16e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−14.06369998352313557055050013786, −12.29198725418011424203322577519, −11.39265132238735796983086913733, −10.66648153908310138482727608471, −9.091440547548671203348794022900, −7.60911093857475668269008381357, −7.01941468976590185527353868349, −4.31812868547547808233943792145, −2.97564220368788261497021050132, −0.54918675981074221086355037558,
1.11436100115256290539758223006, 3.81835754126638731571345322305, 5.39454387300797271260419374057, 7.27335909524070071774159963023, 8.359604497137277982583498809134, 9.100858762053676880000383786534, 10.99859075628963833013699161768, 11.86128024395161070271069971552, 12.95898273165736528996883352787, 14.84855334746025254563774382556
