| L(s) = 1 | + (−16.5 − 13.8i)3-s + (17.0 − 6.21i)5-s + (−92.6 − 160. i)7-s + (38.6 + 219. i)9-s + (107. − 185. i)11-s + (−318. + 266. i)13-s + (−368. − 134. i)15-s + (−270. + 1.53e3i)17-s + (−287. + 1.54e3i)19-s + (−694. + 3.93e3i)21-s + (1.75e3 + 638. i)23-s + (−2.14e3 + 1.79e3i)25-s + (−218. + 377. i)27-s + (−754. − 4.27e3i)29-s + (−2.53e3 − 4.39e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.889i)3-s + (0.305 − 0.111i)5-s + (−0.714 − 1.23i)7-s + (0.159 + 0.902i)9-s + (0.267 − 0.463i)11-s + (−0.521 + 0.438i)13-s + (−0.423 − 0.154i)15-s + (−0.226 + 1.28i)17-s + (−0.182 + 0.983i)19-s + (−0.343 + 1.94i)21-s + (0.691 + 0.251i)23-s + (−0.684 + 0.574i)25-s + (−0.0576 + 0.0997i)27-s + (−0.166 − 0.944i)29-s + (−0.473 − 0.820i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0318504 + 0.0536138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0318504 + 0.0536138i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (287. - 1.54e3i)T \) |
| good | 3 | \( 1 + (16.5 + 13.8i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-17.0 + 6.21i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (92.6 + 160. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-107. + 185. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (318. - 266. i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (270. - 1.53e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-1.75e3 - 638. i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (754. + 4.27e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (2.53e3 + 4.39e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 33.9T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.72e3 - 8.16e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (4.69e3 - 1.70e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (2.24e3 + 1.27e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (2.45e4 + 8.93e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (3.32e3 - 1.88e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-2.96e4 - 1.07e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (5.28e3 + 2.99e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (3.40e4 - 1.24e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (6.28e4 + 5.27e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-8.48e3 - 7.12e3i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-7.46e3 - 1.29e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (7.35e4 - 6.17e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (1.10e4 - 6.27e4i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−12.91691768865339832201226221372, −11.74028216375628945668132808379, −10.72699173241831743957940438288, −9.596498945513725414946290242622, −7.76028766327738370314386936285, −6.65231612586906047918461128192, −5.81714509764541098975873009395, −3.93307136854811046910130188493, −1.46563925202779274220449246504, −0.03122587668022480060479044714,
2.73732812692516364069539341779, 4.76551558557473106183639477879, 5.67255490175197141270746920386, 6.91618792606262348872258162190, 9.050330159142394327835425494981, 9.785753390909060816784748575236, 10.95924641745304549407452024696, 11.96632227322935701469792463417, 12.87133581290796199054841470138, 14.47658196760180080987243080430
