L(s) = 1 | + (1.35 − 2.35i)5-s + (−8.74 + 16.3i)7-s + (−8.39 + 4.84i)11-s − 67.7i·13-s + (50.1 + 86.7i)17-s + (−59.6 − 34.4i)19-s + (−126. − 73.3i)23-s + (58.8 + 101. i)25-s − 284. i·29-s + (197. − 113. i)31-s + (26.5 + 42.7i)35-s + (150. − 261. i)37-s − 232.·41-s + 173.·43-s + (−191. + 331. i)47-s + ⋯ |
L(s) = 1 | + (0.121 − 0.210i)5-s + (−0.471 + 0.881i)7-s + (−0.229 + 0.132i)11-s − 1.44i·13-s + (0.714 + 1.23i)17-s + (−0.720 − 0.416i)19-s + (−1.15 − 0.664i)23-s + (0.470 + 0.814i)25-s − 1.82i·29-s + (1.14 − 0.659i)31-s + (0.128 + 0.206i)35-s + (0.669 − 1.16i)37-s − 0.885·41-s + 0.613·43-s + (−0.594 + 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.085903419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085903419\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (8.74 - 16.3i)T \) |
good | 5 | \( 1 + (-1.35 + 2.35i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (8.39 - 4.84i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 67.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-50.1 - 86.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (59.6 + 34.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (126. + 73.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 284. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-197. + 113. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (191. - 331. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (22.2 - 12.8i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (371. + 644. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (343. + 198. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (293. + 508. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 501. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-616. + 356. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-466. + 807. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 837.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-442. + 766. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.17e3iT - 9.12e5T^{2} \) |
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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
−10.17408826757553859896325094446, −9.481175177568772381733201493790, −8.304518887031029633761270421894, −7.85096814773934549279289337195, −6.20173981149606987172745470432, −5.82182086358739799371118229337, −4.53525491174162834025738466902, −3.21864993324701611502762055444, −2.11953912594285262243678143051, −0.34618329753066434430082904643,
1.30547658313554125066135762741, 2.84627170779629514049792507910, 3.98790655479432617874243898483, 4.99564851081204143336074448283, 6.39578608794965735001766047804, 6.96452347471109430357799336007, 8.001714286136272953030197192842, 9.080308085647208133333061696921, 9.992841160989945380999400750933, 10.56763297379459000829969443204