| L(s) = 1 | + (12 − 20.7i)2-s + (736 + 1.27e3i)4-s + (−2.41e3 − 4.18e3i)5-s + (8.37e3 − 1.45e4i)7-s + 8.44e4·8-s − 1.15e5·10-s + (−2.67e5 + 4.62e5i)11-s + (2.88e5 + 5.00e5i)13-s + (−2.00e5 − 3.48e5i)14-s + (−4.93e5 + 8.54e5i)16-s − 6.90e6·17-s + 1.06e7·19-s + (3.55e6 − 6.15e6i)20-s + (6.41e6 + 1.11e7i)22-s + (−9.32e6 − 1.61e7i)23-s + ⋯ |
| L(s) = 1 | + (0.265 − 0.459i)2-s + (0.359 + 0.622i)4-s + (−0.345 − 0.598i)5-s + (0.188 − 0.326i)7-s + 0.911·8-s − 0.366·10-s + (−0.500 + 0.866i)11-s + (0.215 + 0.373i)13-s + (−0.0998 − 0.172i)14-s + (−0.117 + 0.203i)16-s − 1.17·17-s + 0.987·19-s + (0.248 − 0.430i)20-s + (0.265 + 0.459i)22-s + (−0.301 − 0.523i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.33179 + 1.11751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33179 + 1.11751i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-12 + 20.7i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 5 | \( 1 + (2.41e3 + 4.18e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-8.37e3 + 1.45e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (2.67e5 - 4.62e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-2.88e5 - 5.00e5i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 6.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.06e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (9.32e6 + 1.61e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (6.42e7 - 1.11e8i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-2.64e7 - 4.57e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + 1.82e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (1.54e8 + 2.66e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-8.56e6 + 1.48e7i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (1.34e9 - 2.32e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 1.59e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-2.59e9 - 4.49e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (3.47e9 - 6.02e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-7.74e9 - 1.34e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 9.79e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.46e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.90e10 - 3.30e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-1.46e10 + 2.54e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 + 2.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (3.75e10 - 6.49e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−12.37300186038238821985227056726, −11.40991844607452380534436524616, −10.42029420679697921402836957425, −8.930303086928324127587936346060, −7.79646264163900133062493600863, −6.83384475312086145695832894826, −4.88790164852361810709422631789, −4.03702108009382380596983950359, −2.59268526760733367141118004556, −1.34483357338144171745552410429,
0.38120256459092473850415668244, 1.98537134705752491502212842596, 3.38789624636193712534356595671, 5.05333898197762175545475815796, 6.03398979825719481707148505840, 7.14209488261047329995562885391, 8.239754730501750071293917298355, 9.763433602064005685709958009377, 10.98121130745507295798290310024, 11.51182524412087009359356535827
