L(s) = 1 | + (−2 + 2i)2-s − 13.0i·3-s − 8i·4-s + (−23.5 − 23.5i)5-s + (26.1 + 26.1i)6-s − 19.7·7-s + (16 + 16i)8-s − 90.5·9-s + 94.1·10-s + 152. i·11-s − 104.·12-s + (144. + 144. i)13-s + (39.5 − 39.5i)14-s + (−308. + 308. i)15-s − 64·16-s + (−104. − 104. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s − 1.45i·3-s − 0.5i·4-s + (−0.941 − 0.941i)5-s + (0.727 + 0.727i)6-s − 0.403·7-s + (0.250 + 0.250i)8-s − 1.11·9-s + 0.941·10-s + 1.25i·11-s − 0.727·12-s + (0.853 + 0.853i)13-s + (0.201 − 0.201i)14-s + (−1.37 + 1.37i)15-s − 0.250·16-s + (−0.362 − 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0635284 + 0.247467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0635284 + 0.247467i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 - 2i)T \) |
| 37 | \( 1 + (1.25e3 + 558. i)T \) |
good | 3 | \( 1 + 13.0iT - 81T^{2} \) |
| 5 | \( 1 + (23.5 + 23.5i)T + 625iT^{2} \) |
| 7 | \( 1 + 19.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 152. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-144. - 144. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (104. + 104. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + (227. + 227. i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 + (346. + 346. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (831. - 831. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-237. + 237. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + 316. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.17e3 - 2.17e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 613.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 2.53e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (2.73e3 + 2.73e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (-1.63e3 + 1.63e3i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + 6.63e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.61e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 5.98e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-7.62e3 - 7.62e3i)T + 3.89e7iT^{2} \) |
| 83 | \( 1 + 2.35e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-4.76e3 + 4.76e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (-113. - 113. i)T + 8.85e7iT^{2} \) |
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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
−12.93959476068913057099310694037, −12.38450166396094552814859636147, −11.21157662198762738840076237569, −9.312207425844128359837565861720, −8.315229210924200192662164684726, −7.29919792309983182301431228676, −6.42519270047644667183934474920, −4.50170811890298264778476989304, −1.73178414000045282133590468960, −0.15493793344326275676906982287,
3.28233409351122936408875641431, 3.88202723334510770710677957698, 6.00221715657631957029195891817, 7.88175444510900974664701523884, 8.949448344882931489023996318881, 10.34133776789056636328460518747, 10.81320949544407149509537557456, 11.72056772052007056613310926020, 13.38193186829983313233538468476, 14.80083187139975869977925944634