| L(s) = 1 | + (8.41 + 7.56i)2-s + (27.5 − 11.4i)3-s + (13.5 + 127. i)4-s + (−119. + 289. i)5-s + (318. + 112. i)6-s + (−276. − 276. i)7-s + (−848. + 1.17e3i)8-s + (−915. + 915. i)9-s + (−3.19e3 + 1.52e3i)10-s + (7.84e3 + 3.25e3i)11-s + (1.82e3 + 3.35e3i)12-s + (−2.01e3 − 4.86e3i)13-s + (−234. − 4.41e3i)14-s + 9.34e3i·15-s + (−1.60e4 + 3.44e3i)16-s + 2.66e4i·17-s + ⋯ |
| L(s) = 1 | + (0.743 + 0.668i)2-s + (0.590 − 0.244i)3-s + (0.105 + 0.994i)4-s + (−0.428 + 1.03i)5-s + (0.602 + 0.212i)6-s + (−0.304 − 0.304i)7-s + (−0.586 + 0.810i)8-s + (−0.418 + 0.418i)9-s + (−1.01 + 0.482i)10-s + (1.77 + 0.736i)11-s + (0.305 + 0.560i)12-s + (−0.254 − 0.614i)13-s + (−0.0228 − 0.429i)14-s + 0.714i·15-s + (−0.977 + 0.210i)16-s + 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.40417 + 2.26631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40417 + 2.26631i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8.41 - 7.56i)T \) |
| good | 3 | \( 1 + (-27.5 + 11.4i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (119. - 289. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (276. + 276. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-7.84e3 - 3.25e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (2.01e3 + 4.86e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 2.66e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.19e4 + 2.89e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.49e4 + 4.49e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-1.27e5 + 5.26e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.31e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.89e5 + 4.56e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (5.48e4 - 5.48e4i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-3.12e5 - 1.29e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 2.34e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.03e6 - 4.27e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (7.89e5 - 1.90e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.93e6 - 8.01e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (2.01e6 - 8.32e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.69e6 + 1.69e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-7.48e5 + 7.48e5i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 2.22e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-1.17e6 - 2.84e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-2.52e6 - 2.52e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 6.60e6T + 8.07e13T^{2} \) |
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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
−14.99593942959292087692493835269, −14.74764778869974249817042671393, −13.51248666591753625435510168242, −12.22914782246261579896716051677, −10.83273812035413835714169361359, −8.778793150352425622770136176503, −7.37421078246566754083672494421, −6.41812940495250536888689663194, −4.15003205218778812550752148676, −2.73944275660510132346058098960,
1.01289100436695790713870543468, 3.22154556257159536627477514827, 4.52468070016571072989418137097, 6.31144313554185932269399402925, 8.800471471491751769205445551082, 9.498865741682750250764737235289, 11.64197189760702478144045091579, 12.16159436706639719575409590720, 13.75456704319068953236192501883, 14.55569115440437083184597361450
