| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (−148. + 85.5i)5-s + (−280. + 485. i)7-s + 181. i·8-s − 967.·10-s + (−190. − 110. i)11-s + (−1.89e3 − 3.27e3i)13-s + (−2.74e3 + 1.58e3i)14-s + (−512. + 886. i)16-s + 2.66e3i·17-s + 1.03e4·19-s + (−4.74e3 − 2.73e3i)20-s + (−623. − 1.07e3i)22-s + (1.28e4 − 7.39e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.18 + 0.684i)5-s + (−0.817 + 1.41i)7-s + 0.353i·8-s − 0.967·10-s + (−0.143 − 0.0827i)11-s + (−0.860 − 1.49i)13-s + (−1.00 + 0.578i)14-s + (−0.125 + 0.216i)16-s + 0.541i·17-s + 1.50·19-s + (−0.592 − 0.342i)20-s + (−0.0585 − 0.101i)22-s + (1.05 − 0.607i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1880929368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1880929368\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (148. - 85.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (280. - 485. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (190. + 110. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.89e3 + 3.27e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 2.66e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.03e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.28e4 + 7.39e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.28e4 + 7.43e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (7.91e3 + 1.37e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 2.65e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (9.86e4 - 5.69e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-2.60e4 + 4.51e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.00e4 + 5.77e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.37e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.52e5 - 1.45e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.41e5 + 2.44e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.88e5 + 3.27e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.21e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.11e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-8.77e4 + 1.52e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.14e5 - 1.81e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.23e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.24e5 + 1.08e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.84524874451970898965703824711, −10.71782277494472419235833600908, −9.408918466153809414451457121856, −8.091716144874886483761877887063, −7.31073076492180340304064135378, −6.05289488245290939173318580786, −5.06342798848053632668884851913, −3.34881100185503610228409854652, −2.79391197617246530553070646283, −0.04888604433456453144672460628,
1.16472361840111412673354432846, 3.23739622885667992534780391607, 4.16334128924113114310551018262, 5.06203334429369277586780947868, 7.00816312892898165475363488631, 7.40887995886817567948996383308, 9.140046854190089423256630366604, 10.02114028098999846227624966650, 11.34880546879716571162148329166, 11.91454843011396915399805295438
