| L(s) = 1 | + 13.0i·2-s + 33.4·3-s − 107.·4-s − 173.·5-s + 438. i·6-s − 128.·7-s − 566. i·8-s + 393.·9-s − 2.27e3i·10-s − 408. i·11-s − 3.59e3·12-s − 569. i·13-s − 1.68e3i·14-s − 5.81e3·15-s + 551.·16-s − 7.23e3·17-s + ⋯ |
| L(s) = 1 | + 1.63i·2-s + 1.24·3-s − 1.67·4-s − 1.38·5-s + 2.02i·6-s − 0.375·7-s − 1.10i·8-s + 0.539·9-s − 2.27i·10-s − 0.306i·11-s − 2.08·12-s − 0.259i·13-s − 0.613i·14-s − 1.72·15-s + 0.134·16-s − 1.47·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0491 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0491 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.307641 - 0.323156i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307641 - 0.323156i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-1.00e4 + 2.05e5i)T \) |
| good | 2 | \( 1 - 13.0iT - 64T^{2} \) |
| 3 | \( 1 - 33.4T + 729T^{2} \) |
| 5 | \( 1 + 173.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 128.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 408. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 569. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.23e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.76e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 914. iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.46e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.97e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 8.66e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.53e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.00e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 4.46e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.62e5T + 2.21e10T^{2} \) |
| 61 | \( 1 - 1.21e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 5.10e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.78e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.77e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 8.00e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.02e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.97e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 7.21e5iT - 8.32e11T^{2} \) |
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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
−14.99920247211911651993153499546, −14.01719966236243617463433798081, −13.05622918409674187759929485473, −11.37376917659379392855287714878, −9.322060720599402397494687904782, −8.348980776099983065945710573311, −7.72295823369145535633122002929, −6.54053742973891814219083280277, −4.61791223572179705016859645587, −3.23329235084511355690373788365,
0.15398409746104004235501905848, 2.19493754270576160342253638627, 3.42595195976085650576743702553, 4.27184067261622049602579330542, 7.36713155246206190754414717482, 8.675510375752457601432993585973, 9.494113322504194229148106407236, 10.95206959105165470268985709120, 11.82496576292604926517333402007, 12.89160202789096464709304503445
