| L(s) = 1 | − 5.65i·2-s − 32.0·4-s + 153. i·5-s + 672.·7-s + 181. i·8-s + 868.·10-s + 2.55e3i·11-s − 1.77e3·13-s − 3.80e3i·14-s + 1.02e3·16-s − 3.97e3i·17-s − 7.76e3·19-s − 4.91e3i·20-s + 1.44e4·22-s − 2.61e3i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.22i·5-s + 1.95·7-s + 0.353i·8-s + 0.868·10-s + 1.92i·11-s − 0.805·13-s − 1.38i·14-s + 0.250·16-s − 0.809i·17-s − 1.13·19-s − 0.614i·20-s + 1.35·22-s − 0.214i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.623839454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623839454\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 153. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 672.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.55e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.77e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.97e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.76e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.61e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.44e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.44e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.95e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.63e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.52e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.19e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 5.93e3iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 7.42e3iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 5.45e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.23e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 7.24e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.17e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.72e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.46e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 8.00e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.67e5T + 8.32e11T^{2} \) |
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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
−11.85009723383981435062333210567, −10.93332805146769431312806369303, −10.34193188099466522110229067379, −9.129004956454616027235065956433, −7.70061218915044757987088946809, −7.05346871785274031857988245180, −5.09555344504147403736518570818, −4.30351383789941592818317926919, −2.48180395293667665606355664628, −1.74321159128271946207654877190,
0.46212977243488634455341763186, 1.73262750793074884281652303882, 4.06052551504018568920692979578, 5.08642891170711610027897609980, 5.82503618338148916040609916785, 7.59062003059778755399444506719, 8.552938880836796569442682909156, 8.765623179433449974421770260890, 10.60719276921209661799203019036, 11.50859665434351332886926396081
