| L(s) = 1 | + (2.26 + 11.0i)2-s + (−17.9 + 43.2i)3-s + (−117. + 50.1i)4-s + (−367. + 152. i)5-s + (−519. − 100. i)6-s + (951. − 951. i)7-s + (−822. − 1.19e3i)8-s + (−2.66 − 2.66i)9-s + (−2.52e3 − 3.73e3i)10-s + (2.60e3 + 6.30e3i)11-s + (−59.9 − 5.99e3i)12-s + (−7.36e3 − 3.05e3i)13-s + (1.26e4 + 8.39e3i)14-s − 1.86e4i·15-s + (1.13e4 − 1.18e4i)16-s − 7.29e3i·17-s + ⋯ |
| L(s) = 1 | + (0.199 + 0.979i)2-s + (−0.383 + 0.924i)3-s + (−0.920 + 0.391i)4-s + (−1.31 + 0.544i)5-s + (−0.982 − 0.190i)6-s + (1.04 − 1.04i)7-s + (−0.567 − 0.823i)8-s + (−0.00122 − 0.00122i)9-s + (−0.797 − 1.18i)10-s + (0.591 + 1.42i)11-s + (−0.0100 − 1.00i)12-s + (−0.930 − 0.385i)13-s + (1.23 + 0.817i)14-s − 1.42i·15-s + (0.692 − 0.721i)16-s − 0.360i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0250 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.277038 - 0.270192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277038 - 0.270192i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.26 - 11.0i)T \) |
| good | 3 | \( 1 + (17.9 - 43.2i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (367. - 152. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-951. + 951. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-2.60e3 - 6.30e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (7.36e3 + 3.05e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 7.29e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (2.13e4 + 8.85e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (5.92e4 + 5.92e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (5.13e3 - 1.23e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.50e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (8.48e4 - 3.51e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-5.12e5 - 5.12e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (5.63e4 + 1.36e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.26e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (6.21e5 + 1.50e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (5.25e5 - 2.17e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (3.82e5 - 9.23e5i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.69e5 - 4.09e5i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (5.85e5 - 5.85e5i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.56e6 - 1.56e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 1.87e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (3.81e6 + 1.57e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (8.04e6 - 8.04e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.19e7T + 8.07e13T^{2} \) |
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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
−16.03827330913790329939335054250, −14.93724316422596106807803909545, −14.49240193275168948180327019314, −12.47513536301677556771727787292, −11.08435463789425867680219149054, −9.865139058553375931374467137719, −7.86860377450502734946416206220, −7.10646589739249201540794057583, −4.63838807593233304575805624024, −4.18961703502771259041112873378,
0.18464027956616537829118199141, 1.73032182900263223769383032060, 3.99576372869626572188982889250, 5.66965591367847666147024004018, 7.917082139389251832360656026895, 8.940716866579778436536698045962, 11.24115870448502647705816547432, 11.91889112921326335864645494856, 12.51252162943276572356046238384, 14.11829597466699583052876089309
