| L(s) = 1 | + (12.7 − 9.23i)2-s + (36.7 − 113. i)4-s + (268. + 194. i)5-s + (−72.0 + 221. i)7-s + (44.3 + 136. i)8-s + 5.21e3·10-s + (1.47e3 − 4.15e3i)11-s + (−8.20e3 + 5.96e3i)13-s + (1.13e3 + 3.48e3i)14-s + (1.41e4 + 1.02e4i)16-s + (2.52e4 + 1.83e4i)17-s + (1.47e4 + 4.54e4i)19-s + (3.18e4 − 2.31e4i)20-s + (−1.95e4 − 6.65e4i)22-s + 1.04e4·23-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.816i)2-s + (0.286 − 0.883i)4-s + (0.960 + 0.697i)5-s + (−0.0793 + 0.244i)7-s + (0.0306 + 0.0942i)8-s + 1.64·10-s + (0.335 − 0.942i)11-s + (−1.03 + 0.752i)13-s + (0.110 + 0.339i)14-s + (0.862 + 0.626i)16-s + (1.24 + 0.904i)17-s + (0.494 + 1.52i)19-s + (0.891 − 0.647i)20-s + (−0.392 − 1.33i)22-s + 0.178·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0514i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.41166 - 0.113528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.41166 - 0.113528i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-1.47e3 + 4.15e3i)T \) |
| good | 2 | \( 1 + (-12.7 + 9.23i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-268. - 194. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (72.0 - 221. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (8.20e3 - 5.96e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-2.52e4 - 1.83e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.47e4 - 4.54e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.04e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-3.59e4 + 1.10e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.18e5 + 8.63e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-3.94e4 + 1.21e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (4.68e4 + 1.44e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 3.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.41e5 + 7.41e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.19e6 - 8.67e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-3.21e5 + 9.88e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-3.42e5 - 2.48e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-3.31e6 - 2.41e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-2.42e4 + 7.45e4i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-1.29e6 + 9.39e5i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (7.70e5 + 5.60e5i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 5.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (5.81e6 - 4.22e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.38848293749444292687178722787, −11.73741544498298447979111768127, −10.48752696814496281375232006291, −9.720097294680696478803340767396, −8.027517295405466510711206119472, −6.25390723350289822601540378814, −5.49506428683460430213669493162, −3.88785301124915745975142151281, −2.76137145388483320900680540408, −1.62072477050571754043350991417,
1.03511636381956124150451130021, 2.97062965286703338105569620809, 4.86279045869715078379337622252, 5.19647841066979588345954979041, 6.67234751551351548122602255573, 7.58560346612729354827671655701, 9.391425798354876003210749378610, 10.07916739617154632699832775091, 12.01659530547703644618740631111, 12.82083393586726737840358990031
