| L(s) = 1 | + (−4.74 − 5.66i)2-s + (34.9 − 198. i)4-s + (−15.1 − 41.5i)5-s + (243. + 1.38e3i)7-s + (−2.92e3 + 1.68e3i)8-s + (−163. + 283. i)10-s + (4.42e3 − 1.21e4i)11-s + (4.32e4 + 3.62e4i)13-s + (6.66e3 − 7.94e3i)14-s + (−2.49e4 − 9.09e3i)16-s + (2.65e4 + 1.53e4i)17-s + (6.82e4 + 1.18e5i)19-s + (−8.77e3 + 1.54e3i)20-s + (−8.97e4 + 3.26e4i)22-s + (−4.12e5 − 7.27e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.296 − 0.353i)2-s + (0.136 − 0.774i)4-s + (−0.0242 − 0.0664i)5-s + (0.101 + 0.575i)7-s + (−0.714 + 0.412i)8-s + (−0.0163 + 0.0283i)10-s + (0.302 − 0.829i)11-s + (1.51 + 1.26i)13-s + (0.173 − 0.206i)14-s + (−0.381 − 0.138i)16-s + (0.317 + 0.183i)17-s + (0.523 + 0.906i)19-s + (−0.0548 + 0.00966i)20-s + (−0.383 + 0.139i)22-s + (−1.47 − 0.260i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.68191 - 0.768992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68191 - 0.768992i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (4.74 + 5.66i)T + (-44.4 + 252. i)T^{2} \) |
| 5 | \( 1 + (15.1 + 41.5i)T + (-2.99e5 + 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-243. - 1.38e3i)T + (-5.41e6 + 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-4.42e3 + 1.21e4i)T + (-1.64e8 - 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-4.32e4 - 3.62e4i)T + (1.41e8 + 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-2.65e4 - 1.53e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-6.82e4 - 1.18e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.12e5 + 7.27e4i)T + (7.35e10 + 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-2.50e5 - 2.98e5i)T + (-8.68e10 + 4.92e11i)T^{2} \) |
| 31 | \( 1 + (2.55e4 - 1.44e5i)T + (-8.01e11 - 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-8.98e5 + 1.55e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-2.27e6 + 2.70e6i)T + (-1.38e12 - 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-3.81e5 - 1.38e5i)T + (8.95e12 + 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-4.80e6 + 8.47e5i)T + (2.23e13 - 8.14e12i)T^{2} \) |
| 53 | \( 1 + 1.24e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (3.55e6 + 9.77e6i)T + (-1.12e14 + 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-4.66e5 - 2.64e6i)T + (-1.80e14 + 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-1.30e7 - 1.09e7i)T + (7.05e13 + 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-6.93e6 - 4.00e6i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.28e7 + 2.22e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.24e6 + 1.04e6i)T + (2.63e14 - 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-4.09e7 - 4.87e7i)T + (-3.91e14 + 2.21e15i)T^{2} \) |
| 89 | \( 1 + (2.75e7 - 1.59e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-4.45e7 - 1.62e7i)T + (6.00e15 + 5.03e15i)T^{2} \) |
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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
−12.23535044362979697893747994914, −11.38370800435565837793690010663, −10.44256757204319236397475897134, −9.177052624571255574103790397000, −8.398801339854974229299883685554, −6.42126710199653072623658982535, −5.64869842785008657957317134007, −3.82851151509504262638102106832, −2.06346282816438901950196105105, −0.886435572747332940998375591979,
0.936187772644156042239889112384, 2.97372675544593347402572166383, 4.19859669245610767229059253953, 6.03141267231588112186236877236, 7.30945099294214053784072573353, 8.118225264542351991329123160967, 9.358786640017528761882979422732, 10.65797080613425815959551271174, 11.79892678661785927089600415677, 12.90035096427852567829877992594
