| L(s) = 1 | + (−3.86 − 1.03i)2-s + (13.8 + 7.99i)4-s + (−58.2 + 58.2i)5-s + (−16.8 − 62.7i)7-s + (−45.2 − 45.2i)8-s + (285. − 164. i)10-s + (−30.8 + 115. i)11-s + (403. + 456. i)13-s + 259. i·14-s + (128. + 221. i)16-s + (−805. + 1.39e3i)17-s + (−849. + 227. i)19-s + (−1.27e3 + 341. i)20-s + (238. − 413. i)22-s + (1.26e3 + 2.18e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.433 + 0.249i)4-s + (−1.04 + 1.04i)5-s + (−0.129 − 0.483i)7-s + (−0.249 − 0.249i)8-s + (0.902 − 0.521i)10-s + (−0.0769 + 0.287i)11-s + (0.662 + 0.748i)13-s + 0.354i·14-s + (0.125 + 0.216i)16-s + (−0.676 + 1.17i)17-s + (−0.539 + 0.144i)19-s + (−0.711 + 0.190i)20-s + (0.105 − 0.182i)22-s + (0.497 + 0.861i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03708130030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03708130030\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.86 + 1.03i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-403. - 456. i)T \) |
| good | 5 | \( 1 + (58.2 - 58.2i)T - 3.12e3iT^{2} \) |
| 7 | \( 1 + (16.8 + 62.7i)T + (-1.45e4 + 8.40e3i)T^{2} \) |
| 11 | \( 1 + (30.8 - 115. i)T + (-1.39e5 - 8.05e4i)T^{2} \) |
| 17 | \( 1 + (805. - 1.39e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (849. - 227. i)T + (2.14e6 - 1.23e6i)T^{2} \) |
| 23 | \( 1 + (-1.26e3 - 2.18e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (357. - 206. i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.45e3 + 2.45e3i)T + 2.86e7iT^{2} \) |
| 37 | \( 1 + (-4.73e3 - 1.26e3i)T + (6.00e7 + 3.46e7i)T^{2} \) |
| 41 | \( 1 + (-4.40e3 - 1.18e3i)T + (1.00e8 + 5.79e7i)T^{2} \) |
| 43 | \( 1 + (8.16e3 + 4.71e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.25e4 + 1.25e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 - 7.72e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (3.78e4 - 1.01e4i)T + (6.19e8 - 3.57e8i)T^{2} \) |
| 61 | \( 1 + (-1.20e4 + 2.08e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-9.29e3 + 3.47e4i)T + (-1.16e9 - 6.75e8i)T^{2} \) |
| 71 | \( 1 + (1.07e4 + 4.01e4i)T + (-1.56e9 + 9.02e8i)T^{2} \) |
| 73 | \( 1 + (-2.87e4 + 2.87e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 9.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (9.20e3 - 9.20e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + (-3.20e4 + 1.19e5i)T + (-4.83e9 - 2.79e9i)T^{2} \) |
| 97 | \( 1 + (1.12e5 - 3.00e4i)T + (7.43e9 - 4.29e9i)T^{2} \) |
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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
−11.01027863711464007648310428929, −10.19874351316340890572137750267, −8.965520691124129595721585909766, −7.945092393858218751685597507323, −7.09434664305753196016713387687, −6.28133575689336132861087011769, −4.19790760647661751289194777850, −3.36094883904664622940616398762, −1.79725687937650160946667548885, −0.01669364936505122974200625048,
0.978880873500722408346676494606, 2.81906892670462172755152077184, 4.36502963816642932687218810103, 5.48942455357455484893449949845, 6.80393830605633124093508089721, 8.006944748648437065279930196986, 8.629938207525351318585422620979, 9.381667226749997941695046883577, 10.79525415268611883961707748125, 11.51970814630319391831925096184
