| L(s) = 1 | + (−3.54 − 2.57i)2-s + (3.45 + 10.6i)4-s + (10.9 + 2.29i)5-s − 21.8·7-s + (4.28 − 13.2i)8-s + (−32.8 − 36.2i)10-s + (17.2 + 12.5i)11-s + (−62.3 + 45.2i)13-s + (77.4 + 56.2i)14-s + (23.1 − 16.8i)16-s + (36.3 − 111. i)17-s + (33.1 − 101. i)19-s + (13.3 + 124. i)20-s + (−28.8 − 88.9i)22-s + (82.9 + 60.2i)23-s + ⋯ |
| L(s) = 1 | + (−1.25 − 0.909i)2-s + (0.431 + 1.32i)4-s + (0.978 + 0.205i)5-s − 1.18·7-s + (0.189 − 0.583i)8-s + (−1.03 − 1.14i)10-s + (0.473 + 0.344i)11-s + (−1.32 + 0.966i)13-s + (1.47 + 1.07i)14-s + (0.361 − 0.262i)16-s + (0.519 − 1.59i)17-s + (0.400 − 1.23i)19-s + (0.149 + 1.38i)20-s + (−0.279 − 0.861i)22-s + (0.751 + 0.546i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.223404 - 0.595642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223404 - 0.595642i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-10.9 - 2.29i)T \) |
| good | 2 | \( 1 + (3.54 + 2.57i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 + 21.8T + 343T^{2} \) |
| 11 | \( 1 + (-17.2 - 12.5i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (62.3 - 45.2i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-36.3 + 111. i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-33.1 + 101. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-82.9 - 60.2i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (77.0 + 237. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-44.9 + 138. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (44.0 - 31.9i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-45.9 + 33.3i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 233.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (76.4 + 235. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (46.1 + 141. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-235. + 170. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (181. + 132. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-273. + 841. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-11.8 - 36.5i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (742. + 539. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-283. - 871. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-243. + 750. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-409. - 297. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-5.01 - 15.4i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−11.39010515434002900086608921852, −10.00348436297656288503215536437, −9.463111628314156631949266354158, −9.280200652422792982049234890695, −7.43178891905344259604909776370, −6.66281498143928071521686224291, −5.07517843012129404892448733122, −3.04390005088806491904111750645, −2.14753933030229695546278119242, −0.42768371987317562361916294256,
1.27110869093819400898203433751, 3.25255902631552625735497470739, 5.47078891592071116118228196675, 6.24720982202602053861909698148, 7.16055524037164445884531303745, 8.340709107156077940087632057431, 9.188886784932485183390298978206, 10.11592925161953749426938390072, 10.38897071616989637008272186019, 12.52396365745982388896328212934
