L(s) = 1 | + (199. + 199. i)3-s + (8.93e3 − 8.93e3i)5-s + 5.56e4i·7-s − 9.75e4i·9-s + (1.46e5 − 1.46e5i)11-s + (1.33e6 + 1.33e6i)13-s + 3.56e6·15-s − 2.92e6·17-s + (1.11e7 + 1.11e7i)19-s + (−1.11e7 + 1.11e7i)21-s − 2.12e7i·23-s − 1.10e8i·25-s + (5.48e7 − 5.48e7i)27-s + (−1.31e7 − 1.31e7i)29-s − 1.84e8·31-s + ⋯ |
L(s) = 1 | + (0.474 + 0.474i)3-s + (1.27 − 1.27i)5-s + 1.25i·7-s − 0.550i·9-s + (0.273 − 0.273i)11-s + (0.994 + 0.994i)13-s + 1.21·15-s − 0.499·17-s + (1.03 + 1.03i)19-s + (−0.593 + 0.593i)21-s − 0.688i·23-s − 2.27i·25-s + (0.735 − 0.735i)27-s + (−0.119 − 0.119i)29-s − 1.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(3.50214 + 0.172977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.50214 + 0.172977i\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + (-199. - 199. i)T + 1.77e5iT^{2} \) |
| 5 | \( 1 + (-8.93e3 + 8.93e3i)T - 4.88e7iT^{2} \) |
| 7 | \( 1 - 5.56e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-1.46e5 + 1.46e5i)T - 2.85e11iT^{2} \) |
| 13 | \( 1 + (-1.33e6 - 1.33e6i)T + 1.79e12iT^{2} \) |
| 17 | \( 1 + 2.92e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-1.11e7 - 1.11e7i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + 2.12e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (1.31e7 + 1.31e7i)T + 1.22e16iT^{2} \) |
| 31 | \( 1 + 1.84e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-6.58e7 + 6.58e7i)T - 1.77e17iT^{2} \) |
| 41 | \( 1 - 7.48e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-3.10e8 + 3.10e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.85e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-3.45e9 + 3.45e9i)T - 9.26e18iT^{2} \) |
| 59 | \( 1 + (8.57e8 - 8.57e8i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-6.80e9 - 6.80e9i)T + 4.35e19iT^{2} \) |
| 67 | \( 1 + (-5.85e9 - 5.85e9i)T + 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.15e8iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.23e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 1.66e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-1.21e10 - 1.21e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 3.35e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 2.49e10T + 7.15e21T^{2} \) |
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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
−12.66026011303114673998153021637, −11.70296792461137594353598068310, −9.889959124396437062889384824016, −8.973254935750847427301378424486, −8.699293219318005983802711324887, −6.24743141163517719073813675662, −5.43275090688039206339318912965, −3.96626068630992224150118227545, −2.26849193536035713103441885523, −1.13319646316661544939213737275,
1.12116012468690995717841373459, 2.34826861691037370869564913197, 3.51181290172082127302256916772, 5.50706064873635794458916629231, 6.87771083695532941650828369354, 7.54680063288829697144874314199, 9.260447897184018399820820150833, 10.50550006290405792922044320334, 11.00013101255772626810202583706, 13.19853585122413329507433166356