L(s) = 1 | + (5.70 + 9.88i)5-s + (14.9 − 25.8i)7-s + (−33.1 + 57.3i)11-s + (−19.9 − 34.4i)13-s + 107.·17-s + 70.3·19-s + (−3.45 − 5.99i)23-s + (−2.66 + 4.61i)25-s + (−18.3 + 31.7i)29-s + (115. + 200. i)31-s + 340.·35-s + 36.8·37-s + (−214. − 371. i)41-s + (37.1 − 64.3i)43-s + (−26.2 + 45.5i)47-s + ⋯ |
L(s) = 1 | + (0.510 + 0.884i)5-s + (0.805 − 1.39i)7-s + (−0.907 + 1.57i)11-s + (−0.424 − 0.735i)13-s + 1.53·17-s + 0.849·19-s + (−0.0313 − 0.0543i)23-s + (−0.0213 + 0.0369i)25-s + (−0.117 + 0.203i)29-s + (0.670 + 1.16i)31-s + 1.64·35-s + 0.163·37-s + (−0.817 − 1.41i)41-s + (0.131 − 0.228i)43-s + (−0.0815 + 0.141i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.437397561\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.437397561\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-5.70 - 9.88i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-14.9 + 25.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (33.1 - 57.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (19.9 + 34.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (3.45 + 5.99i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (18.3 - 31.7i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-115. - 200. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 36.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + (214. + 371. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-37.1 + 64.3i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (26.2 - 45.5i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 288.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-391. - 678. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (219. - 380. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-109. - 189. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 790.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (219. - 380. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-25.4 + 44.0i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-599. + 1.03e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.31378526052484551401613349482, −9.803221000780486614011988502846, −8.172249047738657004093302237832, −7.32966936740005056693696897375, −7.08061732797020980394908811383, −5.49724037985137457570934949406, −4.78144034109563403202799788473, −3.49907683495338281909128341131, −2.34842932744882721930734083325, −1.02569938515430774526045928938,
0.893005918541095236221591117660, 2.14474416918060711779626530175, 3.27311616708989915692539026008, 5.01095868415555860512140546012, 5.38468503701235760481462720872, 6.18891970547830072934430100841, 7.921083581727673329895313098580, 8.262929323814184828929056846272, 9.256880861472392411869199212446, 9.844862836053870774517699746931