| L(s) = 1 | + (−2.58 − 3.04i)2-s + (−2.59 + 15.7i)4-s + (5.51 + 9.55i)5-s + (10.3 + 5.95i)7-s + (54.8 − 32.9i)8-s + (14.8 − 41.5i)10-s + (−189. − 109. i)11-s + (18.5 + 32.1i)13-s + (−8.54 − 46.8i)14-s + (−242. − 82.0i)16-s − 284.·17-s + 45.4i·19-s + (−165. + 62.2i)20-s + (157. + 863. i)22-s + (−174. + 100. i)23-s + ⋯ |
| L(s) = 1 | + (−0.647 − 0.762i)2-s + (−0.162 + 0.986i)4-s + (0.220 + 0.382i)5-s + (0.210 + 0.121i)7-s + (0.857 − 0.514i)8-s + (0.148 − 0.415i)10-s + (−1.57 − 0.906i)11-s + (0.109 + 0.189i)13-s + (−0.0435 − 0.239i)14-s + (−0.947 − 0.320i)16-s − 0.982·17-s + 0.126i·19-s + (−0.412 + 0.155i)20-s + (0.325 + 1.78i)22-s + (−0.329 + 0.190i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00275369 + 0.00911195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00275369 + 0.00911195i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.58 + 3.04i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-5.51 - 9.55i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-10.3 - 5.95i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (189. + 109. i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-18.5 - 32.1i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 284.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 45.4iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (174. - 100. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (614. - 1.06e3i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (1.31e3 - 757. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.52e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (1.31e3 + 2.28e3i)T + (-1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (34.6 + 19.9i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-2.49e3 - 1.44e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 1.41e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (2.45e3 - 1.41e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.62e3 + 4.55e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-805. + 465. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 1.16e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.16e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-6.48e3 - 3.74e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-966. - 558. i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 6.73e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (6.02e3 - 1.04e4i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−13.24042367163031214515469467573, −12.27705894787072707905712779606, −10.83641285792725712263342450452, −10.65031990218839899093086188025, −9.109170114192163680208143460916, −8.218836890137430672869365649014, −6.98236437335693020633951613665, −5.23436527619831586681564984472, −3.41464776348280140312454213426, −2.08542621183132975498608919017,
0.00487938322354107417964924550, 2.02278247010574727168099804017, 4.65636480579066515975790976711, 5.67925585228160322109389852031, 7.16886781745421674897470920360, 8.060390976018622801965648698968, 9.225992754996783844050855078291, 10.22454887407854756558965056727, 11.18126422316409246434307857033, 12.88348231721127559053176468993
