| L(s) = 1 | + (−0.769 − 1.33i)2-s + (−15.4 + 2.35i)3-s + (14.8 − 25.6i)4-s + (−8.82 + 15.2i)5-s + (14.9 + 18.7i)6-s + (−82.7 − 143. i)7-s − 94.8·8-s + (231. − 72.4i)9-s + 27.1·10-s + (60.5 + 104. i)11-s + (−167. + 430. i)12-s + (115. − 200. i)13-s + (−127. + 220. i)14-s + (100. − 256. i)15-s + (−401. − 694. i)16-s − 235.·17-s + ⋯ |
| L(s) = 1 | + (−0.136 − 0.235i)2-s + (−0.988 + 0.150i)3-s + (0.462 − 0.801i)4-s + (−0.157 + 0.273i)5-s + (0.170 + 0.212i)6-s + (−0.638 − 1.10i)7-s − 0.524·8-s + (0.954 − 0.298i)9-s + 0.0859·10-s + (0.150 + 0.261i)11-s + (−0.336 + 0.862i)12-s + (0.189 − 0.329i)13-s + (−0.173 + 0.300i)14-s + (0.114 − 0.294i)15-s + (−0.391 − 0.678i)16-s − 0.197·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0330489 + 0.0543113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0330489 + 0.0543113i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (15.4 - 2.35i)T \) |
| 11 | \( 1 + (-60.5 - 104. i)T \) |
| good | 2 | \( 1 + (0.769 + 1.33i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (8.82 - 15.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (82.7 + 143. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 13 | \( 1 + (-115. + 200. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 235.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.09e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.71e3 - 2.97e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.69e3 - 2.92e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (3.86e3 - 6.69e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 6.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.78e3 + 6.56e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-2.48e3 - 4.30e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (2.41e3 + 4.19e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.61e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-6.71e3 + 1.16e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.18e4 + 2.05e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (7.29e3 - 1.26e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.56e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (837. + 1.44e3i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.74e4 - 4.75e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.35e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.95e4 + 1.55e5i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.12180694025598201989257635717, −12.01378724820969812832982233915, −10.83337719319186064608342312531, −10.46287717274132219347635563465, −9.386502083688695684306465679656, −7.22974226794025221202332229639, −6.51095458310828791816962483365, −5.23807471052296200493684838666, −3.63879920223978870625419518157, −1.38134864237283807738377724546,
0.02988255346852996602868368207, 2.37388754701404913036813918957, 4.22407361828843065971659177509, 5.95983683016640112721624303522, 6.63184973877413093337388310010, 8.097518135040080006737599521485, 9.151135114671658402242037346038, 10.66572068058742260301541251458, 11.84416436100573625670348003874, 12.33266985976543749755513476423
