| L(s) = 1 | + (−0.711 − 1.47i)2-s + (20.0 − 1.50i)3-s + (38.2 − 47.9i)4-s + (−40.7 − 132. i)5-s + (−16.5 − 28.6i)6-s + (72.9 + 42.1i)7-s + (−200. − 45.7i)8-s + (−319. + 48.2i)9-s + (−166. + 154. i)10-s + (−897. − 1.12e3i)11-s + (695. − 1.02e3i)12-s + (2.87e3 + 2.66e3i)13-s + (10.3 − 137. i)14-s + (−1.01e3 − 2.59e3i)15-s + (−798. − 3.49e3i)16-s + (−4.39e3 − 1.35e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0889 − 0.184i)2-s + (0.743 − 0.0557i)3-s + (0.597 − 0.748i)4-s + (−0.325 − 1.05i)5-s + (−0.0764 − 0.132i)6-s + (0.212 + 0.122i)7-s + (−0.391 − 0.0893i)8-s + (−0.438 + 0.0661i)9-s + (−0.166 + 0.154i)10-s + (−0.674 − 0.845i)11-s + (0.402 − 0.590i)12-s + (1.30 + 1.21i)13-s + (0.00376 − 0.0502i)14-s + (−0.301 − 0.767i)15-s + (−0.194 − 0.853i)16-s + (−0.894 − 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.15836 - 1.64491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15836 - 1.64491i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.51e4 - 4.55e4i)T \) |
| good | 2 | \( 1 + (0.711 + 1.47i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-20.0 + 1.50i)T + (720. - 108. i)T^{2} \) |
| 5 | \( 1 + (40.7 + 132. i)T + (-1.29e4 + 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-72.9 - 42.1i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (897. + 1.12e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-2.87e3 - 2.66e3i)T + (3.60e5 + 4.81e6i)T^{2} \) |
| 17 | \( 1 + (4.39e3 + 1.35e3i)T + (1.99e7 + 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-1.46e3 + 9.73e3i)T + (-4.49e7 - 1.38e7i)T^{2} \) |
| 23 | \( 1 + (-1.54e3 + 3.93e3i)T + (-1.08e8 - 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-4.18e4 - 3.13e3i)T + (5.88e8 + 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-3.84e4 - 2.62e4i)T + (3.24e8 + 8.26e8i)T^{2} \) |
| 37 | \( 1 + (4.51e4 - 2.60e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-8.81e4 + 4.24e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (8.43e4 - 1.05e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-3.70e4 + 3.44e4i)T + (1.65e9 - 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-1.68e4 - 7.38e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.25e5 - 1.84e5i)T + (-1.88e10 + 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-3.49e4 - 5.26e3i)T + (8.64e10 + 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-3.12e5 + 1.22e5i)T + (9.39e10 - 8.71e10i)T^{2} \) |
| 73 | \( 1 + (2.19e5 - 2.36e5i)T + (-1.13e10 - 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-3.46e5 + 6.00e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.23e4 - 1.64e5i)T + (-3.23e11 + 4.87e10i)T^{2} \) |
| 89 | \( 1 + (1.64e5 - 1.23e4i)T + (4.91e11 - 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-5.51e5 - 6.91e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.18497944494726463052411176109, −13.41813700812448372040172049473, −11.73749875261327568165731625848, −10.88681952464348963084900690193, −9.013945412560049865609900114965, −8.472427178281014422910554377998, −6.48461006139524812611145249565, −4.84024229756012465560842385512, −2.68169523108070296947584126214, −0.914956939658001680780740714456,
2.54768297628614550060868235009, 3.61701064782658833710619376704, 6.24830420244785909107429800621, 7.70488288335900505128028415174, 8.378133545826159214487909782595, 10.35612450950777960675309422916, 11.35970372650755012025723842365, 12.75388568185062464855265741352, 14.05880306259033760028586928654, 15.26939835571930475127832630367
