| 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
−12.90035096427852567829877992594, −11.79892678661785927089600415677, −10.65797080613425815959551271174, −9.358786640017528761882979422732, −8.118225264542351991329123160967, −7.30945099294214053784072573353, −6.03141267231588112186236877236, −4.19859669245610767229059253953, −2.97372675544593347402572166383, −0.936187772644156042239889112384,
0.886435572747332940998375591979, 2.06346282816438901950196105105, 3.82851151509504262638102106832, 5.64869842785008657957317134007, 6.42126710199653072623658982535, 8.398801339854974229299883685554, 9.177052624571255574103790397000, 10.44256757204319236397475897134, 11.38370800435565837793690010663, 12.23535044362979697893747994914
