| L(s) = 1 | + (−0.145 + 1.40i)2-s + (−0.0980 + 0.995i)3-s + (−1.95 − 0.408i)4-s + (3.58 − 1.91i)5-s + (−1.38 − 0.282i)6-s + (−0.464 − 2.33i)7-s + (0.859 − 2.69i)8-s + (−0.980 − 0.195i)9-s + (2.17 + 5.32i)10-s + (2.45 − 2.99i)11-s + (0.598 − 1.90i)12-s + (0.816 − 1.52i)13-s + (3.35 − 0.313i)14-s + (1.55 + 3.75i)15-s + (3.66 + 1.60i)16-s + (0.917 − 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.102 + 0.994i)2-s + (−0.0565 + 0.574i)3-s + (−0.978 − 0.204i)4-s + (1.60 − 0.857i)5-s + (−0.565 − 0.115i)6-s + (−0.175 − 0.882i)7-s + (0.303 − 0.952i)8-s + (−0.326 − 0.0650i)9-s + (0.688 + 1.68i)10-s + (0.740 − 0.902i)11-s + (0.172 − 0.550i)12-s + (0.226 − 0.423i)13-s + (0.895 − 0.0839i)14-s + (0.402 + 0.970i)15-s + (0.916 + 0.400i)16-s + (0.222 − 0.537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41201 + 0.365516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41201 + 0.365516i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.145 - 1.40i)T \) |
| 3 | \( 1 + (0.0980 - 0.995i)T \) |
| good | 5 | \( 1 + (-3.58 + 1.91i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.464 + 2.33i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.45 + 2.99i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.816 + 1.52i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.917 + 2.21i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.208 + 0.687i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (6.95 - 4.64i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.23 - 4.29i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (5.24 - 5.24i)T - 31iT^{2} \) |
| 37 | \( 1 + (-8.27 - 2.50i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.81 - 5.71i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.834 - 8.47i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-2.77 - 1.14i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (5.60 + 4.59i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (0.745 + 1.39i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.755 + 0.0744i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 0.124i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-0.459 + 0.0913i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.858 - 4.31i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-14.1 + 5.84i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (9.33 - 2.83i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-2.54 - 1.69i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-6.99 + 6.99i)T - 97iT^{2} \) |
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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
−11.14741215250303621186795916316, −10.04820740421267644021407703579, −9.503955858641716400890251804831, −8.836076360470116277026024827622, −7.72049007449127727821732760238, −6.33835603173212434836881934727, −5.74732092652322594280192752773, −4.82134496882860378194314943044, −3.60606442530445574147673520891, −1.16480901286103203455673341174,
1.95857411559617663341451940682, 2.31651007948835743393079597660, 3.99101676023949447649329032583, 5.71259542009268790693110874470, 6.20086475934330106443336528856, 7.54580875800274570869950374300, 8.969520037399959989886752630228, 9.530331477716198308715686558741, 10.32109742477694250017061891904, 11.25067124870687506628371769264
