| L(s) = 1 | + (11.3 − 0.356i)2-s + (27.3 + 66.0i)3-s + (127. − 8.05i)4-s + (250. + 103. i)5-s + (332. + 737. i)6-s + (−662. − 662. i)7-s + (1.44e3 − 136. i)8-s + (−2.06e3 + 2.06e3i)9-s + (2.87e3 + 1.08e3i)10-s + (12.1 − 29.3i)11-s + (4.02e3 + 8.21e3i)12-s + (−7.27e3 + 3.01e3i)13-s + (−7.72e3 − 7.25e3i)14-s + 1.93e4i·15-s + (1.62e4 − 2.05e3i)16-s − 1.47e4i·17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0314i)2-s + (0.584 + 1.41i)3-s + (0.998 − 0.0629i)4-s + (0.896 + 0.371i)5-s + (0.629 + 1.39i)6-s + (−0.729 − 0.729i)7-s + (0.995 − 0.0943i)8-s + (−0.945 + 0.945i)9-s + (0.907 + 0.342i)10-s + (0.00275 − 0.00664i)11-s + (0.672 + 1.37i)12-s + (−0.918 + 0.380i)13-s + (−0.752 − 0.706i)14-s + 1.48i·15-s + (0.992 − 0.125i)16-s − 0.726i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.32949 + 2.13219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.32949 + 2.13219i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-11.3 + 0.356i)T \) |
| good | 3 | \( 1 + (-27.3 - 66.0i)T + (-1.54e3 + 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-250. - 103. i)T + (5.52e4 + 5.52e4i)T^{2} \) |
| 7 | \( 1 + (662. + 662. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-12.1 + 29.3i)T + (-1.37e7 - 1.37e7i)T^{2} \) |
| 13 | \( 1 + (7.27e3 - 3.01e3i)T + (4.43e7 - 4.43e7i)T^{2} \) |
| 17 | \( 1 + 1.47e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-4.64e4 + 1.92e4i)T + (6.32e8 - 6.32e8i)T^{2} \) |
| 23 | \( 1 + (6.78e4 - 6.78e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (5.75e4 + 1.38e5i)T + (-1.21e10 + 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (3.66e5 + 1.51e5i)T + (6.71e10 + 6.71e10i)T^{2} \) |
| 41 | \( 1 + (1.66e5 - 1.66e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-3.15e5 + 7.62e5i)T + (-1.92e11 - 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.19e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (2.08e5 - 5.02e5i)T + (-8.30e11 - 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.20e6 - 4.99e5i)T + (1.75e12 + 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-5.31e5 - 1.28e6i)T + (-2.22e12 + 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.29e6 - 3.12e6i)T + (-4.28e12 + 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.40e6 + 1.40e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-9.44e5 + 9.44e5i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 2.52e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (8.11e6 - 3.36e6i)T + (1.91e13 - 1.91e13i)T^{2} \) |
| 89 | \( 1 + (4.96e6 + 4.96e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.10e7T + 8.07e13T^{2} \) |
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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
−15.44103098995990064199312215307, −13.97461619766189482486125108992, −13.75054490205807750062196586476, −11.72089536635436974797344182137, −10.10195687676007784709212282590, −9.698343170291065976040346566730, −7.16998472188719948473515748057, −5.42710241380221905983118970942, −3.96457243928977119108995419245, −2.67502382285346892855535757062,
1.67439710170534291453282925602, 2.89406199907786041254684015258, 5.55109156993170681597078219367, 6.65761734310604446712027817093, 8.092098040075933070052283662721, 9.879417213785759783196543469521, 12.17871560539241194589882441240, 12.64860521603777476169684627872, 13.69392254318419178253664133988, 14.52640274844390499382563425628
