L(s) = 1 | + (0.701 − 3.93i)2-s + (−15.0 − 5.52i)4-s + (−14.3 − 24.7i)5-s + (−22.2 − 12.8i)7-s + (−32.3 + 55.2i)8-s + (−107. + 38.9i)10-s + (93.9 + 54.2i)11-s + (44.2 + 76.5i)13-s + (−66.0 + 78.4i)14-s + (194. + 165. i)16-s − 504.·17-s + 191. i·19-s + (77.8 + 451. i)20-s + (279. − 331. i)22-s + (−831. + 480. i)23-s + ⋯ |
L(s) = 1 | + (0.175 − 0.984i)2-s + (−0.938 − 0.345i)4-s + (−0.572 − 0.991i)5-s + (−0.453 − 0.261i)7-s + (−0.504 + 0.863i)8-s + (−1.07 + 0.389i)10-s + (0.776 + 0.448i)11-s + (0.261 + 0.453i)13-s + (−0.337 + 0.400i)14-s + (0.761 + 0.648i)16-s − 1.74·17-s + 0.530i·19-s + (0.194 + 1.12i)20-s + (0.577 − 0.685i)22-s + (−1.57 + 0.907i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.139790 + 0.168314i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.139790 + 0.168314i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.701 + 3.93i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (14.3 + 24.7i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (22.2 + 12.8i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-93.9 - 54.2i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-44.2 - 76.5i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 504.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 191. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (831. - 480. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-396. + 687. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-285. + 164. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 209.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-528. - 914. i)T + (-1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (2.88e3 + 1.66e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (977. + 564. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.13e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (4.03e3 - 2.33e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.79e3 + 4.84e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (6.12e3 - 3.53e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.95e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.52e3 - 879. i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-2.62e3 - 1.51e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 559.T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.10e3 + 1.90e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−12.07050800801645281411858168959, −11.51893816014971735831538420611, −10.07520817952019373999098888474, −9.147840568788194743042155956944, −8.177133908600341714047542349121, −6.34509201166847510816964558119, −4.61443298516899598425969773394, −3.84214522786289751743099056486, −1.77013396024763557364279519874, −0.091755453143157531526890363833,
3.12592385104325366636172917194, 4.41327574590679884069335417924, 6.23462757260635748677975838863, 6.80579846806749033467924362323, 8.168808482905231260156751077982, 9.151876430532316788380858343077, 10.56032893835853606572787909866, 11.71850683147113188448389967986, 12.93584677994051867063590338702, 13.96598820414350051289917277309