| L(s) = 1 | + (−3.07 − 1.48i)2-s + (18.8 − 23.6i)3-s + (−12.6 − 15.8i)4-s + (15.8 + 7.61i)5-s + (−93.1 + 44.8i)6-s + (−39.8 + 49.9i)7-s + (39.7 + 174. i)8-s + (−149. − 656. i)9-s + (−37.3 − 46.8i)10-s + (−25.0 + 109. i)11-s − 615.·12-s + (148. − 650. i)13-s + (196. − 94.6i)14-s + (478. − 230. i)15-s + (−8.84 + 38.7i)16-s − 137.·17-s + ⋯ |
| L(s) = 1 | + (−0.544 − 0.262i)2-s + (1.21 − 1.51i)3-s + (−0.396 − 0.496i)4-s + (0.282 + 0.136i)5-s + (−1.05 + 0.508i)6-s + (−0.307 + 0.385i)7-s + (0.219 + 0.962i)8-s + (−0.616 − 2.70i)9-s + (−0.118 − 0.148i)10-s + (−0.0625 + 0.273i)11-s − 1.23·12-s + (0.243 − 1.06i)13-s + (0.268 − 0.129i)14-s + (0.549 − 0.264i)15-s + (−0.00864 + 0.0378i)16-s − 0.115·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.585030 - 1.32448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585030 - 1.32448i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-4.34e3 + 1.28e3i)T \) |
| good | 2 | \( 1 + (3.07 + 1.48i)T + (19.9 + 25.0i)T^{2} \) |
| 3 | \( 1 + (-18.8 + 23.6i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-15.8 - 7.61i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (39.8 - 49.9i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (25.0 - 109. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-148. + 650. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 137.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.46e3 - 1.84e3i)T + (-5.50e5 + 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-2.89e3 + 1.39e3i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (-76.5 - 36.8i)T + (1.78e7 + 2.23e7i)T^{2} \) |
| 37 | \( 1 + (-2.81e3 - 1.23e4i)T + (-6.24e7 + 3.00e7i)T^{2} \) |
| 41 | \( 1 - 6.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-6.40e3 + 3.08e3i)T + (9.16e7 - 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-3.65e3 + 1.60e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (7.97e3 + 3.84e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (3.13e4 - 3.93e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-8.06e3 - 3.53e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (1.76e3 - 7.73e3i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (4.37e4 - 2.10e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + (7.77e3 + 3.40e4i)T + (-2.77e9 + 1.33e9i)T^{2} \) |
| 83 | \( 1 + (-4.42e4 - 5.54e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.48e4 - 3.12e4i)T + (3.48e9 + 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-1.28e4 - 1.61e4i)T + (-1.91e9 + 8.37e9i)T^{2} \) |
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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
−15.20120541750708167156132525294, −14.18122772991935554867815894367, −13.28646941075663159220757440184, −12.12735693500632428315086201074, −10.05072606621490370656719644408, −8.821543064412819862002275942472, −7.80031268273132826986343821256, −6.08053547120495296966373808278, −2.72520907309960657911144243960, −1.09574295131732243167158383902,
3.26924425271162849735175333915, 4.61982269397124780825804873542, 7.55680388639523281150594918673, 9.135644955393102090323031563669, 9.359972493861369464406088333702, 10.92972129613059298899146173764, 13.34716553376748750164983645280, 14.08768686292340910533022305758, 15.64265226134695821959135691956, 16.31810494523229425033209348107
