| L(s) = 1 | + (−1.53 − 0.781i)2-s + (−1.71 − 0.270i)3-s + (−0.608 − 0.837i)4-s + (0.173 + 4.99i)5-s + (2.41 + 1.75i)6-s + (−1.91 + 1.91i)7-s + (1.35 + 8.56i)8-s + (2.85 + 0.927i)9-s + (3.63 − 7.80i)10-s + (4.45 + 13.6i)11-s + (0.814 + 1.59i)12-s + (−12.0 + 6.14i)13-s + (4.43 − 1.44i)14-s + (1.05 − 8.59i)15-s + (3.33 − 10.2i)16-s + (−16.9 + 2.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.767 − 0.390i)2-s + (−0.570 − 0.0903i)3-s + (−0.152 − 0.209i)4-s + (0.0347 + 0.999i)5-s + (0.402 + 0.292i)6-s + (−0.273 + 0.273i)7-s + (0.169 + 1.07i)8-s + (0.317 + 0.103i)9-s + (0.363 − 0.780i)10-s + (0.404 + 1.24i)11-s + (0.0678 + 0.133i)12-s + (−0.927 + 0.472i)13-s + (0.316 − 0.102i)14-s + (0.0704 − 0.573i)15-s + (0.208 − 0.641i)16-s + (−0.994 + 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.306081 + 0.298602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.306081 + 0.298602i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.270i)T \) |
| 5 | \( 1 + (-0.173 - 4.99i)T \) |
| good | 2 | \( 1 + (1.53 + 0.781i)T + (2.35 + 3.23i)T^{2} \) |
| 7 | \( 1 + (1.91 - 1.91i)T - 49iT^{2} \) |
| 11 | \( 1 + (-4.45 - 13.6i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (12.0 - 6.14i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (16.9 - 2.67i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-10.4 + 14.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (6.25 - 12.2i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (19.8 + 27.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-17.3 - 12.5i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-2.16 - 4.24i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (24.9 - 76.8i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-37.2 - 37.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.63 + 48.2i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-52.4 - 8.30i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-23.5 - 7.65i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (28.2 + 87.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 1.82i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (11.5 - 8.38i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-54.4 + 106. i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-38.1 - 52.4i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (14.1 + 89.4i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (21.2 - 6.91i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (9.25 - 58.4i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.69635592734358383792933607928, −13.48042691236427391628899571785, −11.94540205684678326725828485978, −11.17868812492761949978241669704, −9.970587368101767426475928665520, −9.384291215685885369338187256323, −7.53219567833436391694268999003, −6.39465872971254692723249703875, −4.72244484873506054977251106978, −2.20808806960996701815975293694,
0.50047219493750024417285361902, 3.99857779992864228104001230071, 5.59273778957145040370204182746, 7.11345408349937679687657440900, 8.406627481291312004447489144788, 9.294466538391707232339811442119, 10.43772362362220789846710280930, 11.93747545537698889967951133828, 12.83849843043676827469392952938, 13.86036179103279743366412219280
