| L(s) = 1 | + (−24.7 − 31.0i)2-s + (98.3 + 14.8i)3-s + (−236. + 1.03e3i)4-s + (1.49e3 + 1.01e3i)5-s + (−1.97e3 − 3.41e3i)6-s + (2.56e3 − 4.44e3i)7-s + (1.96e4 − 9.47e3i)8-s + (−9.35e3 − 2.88e3i)9-s + (−5.36e3 − 7.15e4i)10-s + (1.42e4 + 6.25e4i)11-s + (−3.86e4 + 9.83e4i)12-s + (−3.96e3 + 5.28e4i)13-s + (−2.01e5 + 3.03e4i)14-s + (1.31e5 + 1.22e5i)15-s + (−2.90e5 − 1.40e5i)16-s + (−4.80e5 + 3.27e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 − 1.37i)2-s + (0.701 + 0.105i)3-s + (−0.461 + 2.02i)4-s + (1.06 + 0.728i)5-s + (−0.621 − 1.07i)6-s + (0.403 − 0.699i)7-s + (1.69 − 0.818i)8-s + (−0.475 − 0.146i)9-s + (−0.169 − 2.26i)10-s + (0.294 + 1.28i)11-s + (−0.537 + 1.36i)12-s + (−0.0384 + 0.513i)13-s + (−1.40 + 0.211i)14-s + (0.672 + 0.623i)15-s + (−1.10 − 0.534i)16-s + (−1.39 + 0.950i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.08994 + 0.323706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08994 + 0.323706i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.35e7 + 1.78e7i)T \) |
| good | 2 | \( 1 + (24.7 + 31.0i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (-98.3 - 14.8i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-1.49e3 - 1.01e3i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-2.56e3 + 4.44e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.42e4 - 6.25e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (3.96e3 - 5.28e4i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (4.80e5 - 3.27e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (2.91e5 - 9.00e4i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (-2.59e4 + 2.41e4i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (1.11e4 - 1.67e3i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (2.80e6 - 7.15e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (-2.86e6 - 4.96e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.77e7 - 2.22e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (-7.73e6 + 3.39e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-9.28e5 - 1.23e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-4.55e6 - 2.19e6i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (1.35e7 + 3.46e7i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (-1.90e8 + 5.87e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-3.01e8 - 2.79e8i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (2.93e7 - 3.91e8i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (1.98e7 - 3.43e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.39e8 + 2.10e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (-4.55e8 - 6.87e7i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-6.24e7 - 2.73e8i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−13.93859413092322297787536160032, −12.72551703768958225634182764090, −11.24093252470722704344911224376, −10.33266670401073730446984008963, −9.449517752212814629105810734553, −8.414411485092618887220303174565, −6.80982839117436184934400271997, −4.02426200131222237588397449061, −2.44344041025768899584068028616, −1.68270205150790735420316683861,
0.49941747637462772197986793419, 2.22521967721666521233641706615, 5.31911012417849587310885672867, 6.17202961827634143662684939467, 7.928302901559647655598868013908, 8.940046965714371212000011919661, 9.246773396412002695401835593588, 11.08664801235637748278070658415, 13.29103496678590275467421369507, 14.14006132288033189105960053417
