| L(s) = 1 | + 8.37·2-s + 6.18·4-s + (−60.3 − 109. i)5-s − 270. i·7-s − 484.·8-s + (−505. − 917. i)10-s − 1.18e3i·11-s − 526. i·13-s − 2.26e3i·14-s − 4.45e3·16-s + 6.58e3·17-s − 9.08e3·19-s + (−373. − 677. i)20-s − 9.94e3i·22-s + 8.92e3·23-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0967·4-s + (−0.482 − 0.875i)5-s − 0.787i·7-s − 0.945·8-s + (−0.505 − 0.917i)10-s − 0.891i·11-s − 0.239i·13-s − 0.824i·14-s − 1.08·16-s + 1.34·17-s − 1.32·19-s + (−0.0466 − 0.0847i)20-s − 0.934i·22-s + 0.733·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.959389 - 1.53172i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.959389 - 1.53172i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (60.3 + 109. i)T \) |
| good | 2 | \( 1 - 8.37T + 64T^{2} \) |
| 7 | \( 1 + 270. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.18e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 526. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.58e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.08e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 8.92e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 7.73e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.11e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.59e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 5.67e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.26e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.82e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 3.18e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.19e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.59e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.87e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.70e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.23e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.60e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 7.40e3T + 3.26e11T^{2} \) |
| 89 | \( 1 + 9.91e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.52e6iT - 8.32e11T^{2} \) |
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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
−14.03368973003921390877976869609, −13.06037935136067182127899873443, −12.24736099993751120684968199495, −10.88918997881441639780944081325, −9.159191792494955053643867426004, −7.87273135904628673360055431523, −5.96057966843679878323316994321, −4.62833247453272525057936951865, −3.47392080392460841180517067104, −0.60867861073591449190105460913,
2.67744495647792951729532820976, 4.13394851637830763759365721559, 5.64283405797301429322764247889, 7.04131169782325112854075847643, 8.759028497958050315413120030897, 10.31441678784652386724029503069, 11.86849290704495385697700188165, 12.51610884733125554002086884805, 13.93431839407139444262578283895, 14.97337475713652269308436659703
