L(s) = 1 | + (296. + 296. i)3-s + (−2.86e3 + 2.86e3i)5-s − 8.38e3i·7-s − 1.24e3i·9-s + (−8.45e4 + 8.45e4i)11-s + (−3.85e5 − 3.85e5i)13-s − 1.70e6·15-s + 1.56e6·17-s + (−9.59e6 − 9.59e6i)19-s + (2.48e6 − 2.48e6i)21-s − 3.88e7i·23-s + 3.23e7i·25-s + (5.29e7 − 5.29e7i)27-s + (−8.81e7 − 8.81e7i)29-s + 7.23e7·31-s + ⋯ |
L(s) = 1 | + (0.704 + 0.704i)3-s + (−0.410 + 0.410i)5-s − 0.188i·7-s − 0.00701i·9-s + (−0.158 + 0.158i)11-s + (−0.287 − 0.287i)13-s − 0.578·15-s + 0.267·17-s + (−0.888 − 0.888i)19-s + (0.132 − 0.132i)21-s − 1.25i·23-s + 0.663i·25-s + (0.709 − 0.709i)27-s + (−0.797 − 0.797i)29-s + 0.453·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(1.26570 - 0.827796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26570 - 0.827796i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-296. - 296. i)T + 1.77e5iT^{2} \) |
| 5 | \( 1 + (2.86e3 - 2.86e3i)T - 4.88e7iT^{2} \) |
| 7 | \( 1 + 8.38e3iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (8.45e4 - 8.45e4i)T - 2.85e11iT^{2} \) |
| 13 | \( 1 + (3.85e5 + 3.85e5i)T + 1.79e12iT^{2} \) |
| 17 | \( 1 - 1.56e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (9.59e6 + 9.59e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + 3.88e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (8.81e7 + 8.81e7i)T + 1.22e16iT^{2} \) |
| 31 | \( 1 - 7.23e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (9.98e7 - 9.98e7i)T - 1.77e17iT^{2} \) |
| 41 | \( 1 - 1.13e9iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.05e9 + 1.05e9i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.33e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-2.50e9 + 2.50e9i)T - 9.26e18iT^{2} \) |
| 59 | \( 1 + (-3.81e9 + 3.81e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (7.73e9 + 7.73e9i)T + 4.35e19iT^{2} \) |
| 67 | \( 1 + (2.91e9 + 2.91e9i)T + 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.77e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.04e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.60e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (5.73e9 + 5.73e9i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 - 1.07e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 7.36e9T + 7.15e21T^{2} \) |
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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
−12.41315466060794264066728289549, −11.07030413975326758688302695542, −10.06701577010146497092921366665, −8.984859741863642695208819488961, −7.84207740564751903284680122132, −6.52099693385353101556403009854, −4.71241397047389664617021480786, −3.59613330345259582365347177046, −2.45482422511909965405798379614, −0.37148360034496407499428693523,
1.31154799122229639243240843770, 2.51053350680642274359220835102, 4.01462260328181160322038463312, 5.59048748937640380755661712123, 7.20352721666072479968700730634, 8.111317584663173904547520137993, 9.057683876581734189802783229065, 10.55284432547660755038368145800, 11.97153411634475275117125714720, 12.81775969074826862487137452786