| L(s) = 1 | + (3.23 + 2.35i)2-s + (2.78 + 8.55i)3-s + (4.94 + 15.2i)4-s + (−21.1 + 51.7i)5-s + (−11.1 + 34.2i)6-s + 92.5·7-s + (−19.7 + 60.8i)8-s + (−65.5 + 47.6i)9-s + (−190. + 117. i)10-s + (377. + 274. i)11-s + (−116. + 84.6i)12-s + (−257. + 187. i)13-s + (299. + 217. i)14-s + (−501. − 36.8i)15-s + (−207. + 150. i)16-s + (27.7 − 85.5i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.377 + 0.925i)5-s + (−0.126 + 0.388i)6-s + 0.714·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (−0.600 + 0.372i)10-s + (0.940 + 0.683i)11-s + (−0.233 + 0.169i)12-s + (−0.422 + 0.307i)13-s + (0.408 + 0.296i)14-s + (−0.575 − 0.0423i)15-s + (−0.202 + 0.146i)16-s + (0.0233 − 0.0718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.444935375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444935375\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 5 | \( 1 + (21.1 - 51.7i)T \) |
| good | 7 | \( 1 - 92.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-377. - 274. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (257. - 187. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-27.7 + 85.5i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (361. - 1.11e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.86e3 + 1.35e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.47e3 + 4.54e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.74e3 - 5.36e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (691. - 502. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-9.80e3 + 7.12e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.04e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.46e3 - 1.68e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-58.8 - 181. i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (5.23e3 - 3.80e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.64e3 - 2.64e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-3.00e3 + 9.24e3i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-5.06e3 - 1.55e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.62e4 - 3.36e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.30e4 - 4.00e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.62e4 + 5.01e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-5.14e4 - 3.73e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-2.54e4 - 7.82e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.42273751221826217495864012489, −11.63937502104500828065727491821, −10.66121225267718440226692276777, −9.531197404896221208381700495948, −8.176905845026589844753538019658, −7.21916753931698439245128431688, −6.09932214717881285494340125208, −4.58965401479952852132088352263, −3.74988775725062912150085470250, −2.18915755917880858009153089397,
0.66326310107304404171473116396, 1.87093699280759030560895337508, 3.61417976644315035584117388753, 4.81999815095408926610485930773, 5.97435966110842002188776252282, 7.44167411954464307244699840690, 8.510032986472553508685593666894, 9.464572912815639090131415320401, 11.09204824334686430304072007100, 11.76163059215100986924719003528
