| L(s) = 1 | + (−4.53 + 10.3i)2-s + (−44.2 + 15.1i)3-s + (−86.9 − 93.9i)4-s + (34.4 + 59.7i)5-s + (44.0 − 527. i)6-s + (−1.42e3 − 822. i)7-s + (1.36e3 − 474. i)8-s + (1.73e3 − 1.33e3i)9-s + (−775. + 86.7i)10-s + (1.67e3 + 965. i)11-s + (5.26e3 + 2.84e3i)12-s + (−494. + 285. i)13-s + (1.49e4 − 1.10e4i)14-s + (−2.42e3 − 2.12e3i)15-s + (−1.27e3 + 1.63e4i)16-s − 2.70e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.400 + 0.916i)2-s + (−0.946 + 0.323i)3-s + (−0.678 − 0.734i)4-s + (0.123 + 0.213i)5-s + (0.0831 − 0.996i)6-s + (−1.56 − 0.906i)7-s + (0.944 − 0.327i)8-s + (0.791 − 0.611i)9-s + (−0.245 + 0.0274i)10-s + (0.378 + 0.218i)11-s + (0.879 + 0.475i)12-s + (−0.0623 + 0.0360i)13-s + (1.45 − 1.07i)14-s + (−0.185 − 0.162i)15-s + (−0.0779 + 0.996i)16-s − 1.33i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.323163 + 0.435448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.323163 + 0.435448i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.53 - 10.3i)T \) |
| 3 | \( 1 + (44.2 - 15.1i)T \) |
| good | 5 | \( 1 + (-34.4 - 59.7i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (1.42e3 + 822. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.67e3 - 965. i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (494. - 285. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 2.70e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.73e4 - 4.73e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (5.57e4 - 9.65e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (2.10e5 - 1.21e5i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.66e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + (3.73e5 - 2.15e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.74e5 + 3.02e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.90e5 - 6.76e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.33e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.13e6 + 1.23e6i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.99e6 - 1.14e6i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.23e5 - 5.59e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-2.86e6 - 1.65e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.79e6 - 1.03e6i)T + (1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 9.30e5iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (7.52e6 - 1.30e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.57201460404255582676008904445, −12.59134562629241423292445305371, −10.92830169673111643993443727161, −9.968900801298332389018404634866, −9.232005307972504575631704101362, −7.06860282378587629586773948640, −6.67534066695013067717427099113, −5.26663203438077447161607882511, −3.79944611744033038659087484104, −0.74250741452344226767889506552,
0.39304816540778880040564320440, 2.09002185855313832805944533463, 3.77824212300577674169388799813, 5.58501469364023139050752534282, 6.76859128937631544741726062732, 8.570745074348593457836828474572, 9.647207540697731942926442056026, 10.67142558261950336073159413157, 11.81666205332601418680665333654, 12.85812464400715983987164217756
