| L(s) = 1 | + (3.99 + 0.138i)2-s + (8.69 + 3.97i)3-s + (15.9 + 1.10i)4-s + (17.8 − 20.5i)5-s + (34.2 + 17.0i)6-s + (0.355 + 0.553i)7-s + (63.6 + 6.62i)8-s + (6.83 + 7.88i)9-s + (74.1 − 79.8i)10-s + (−177. − 25.5i)11-s + (134. + 73.0i)12-s + (85.7 + 55.1i)13-s + (1.34 + 2.26i)14-s + (236. − 108. i)15-s + (253. + 35.2i)16-s + (−456. − 134. i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0345i)2-s + (0.966 + 0.441i)3-s + (0.997 + 0.0690i)4-s + (0.713 − 0.823i)5-s + (0.950 + 0.474i)6-s + (0.00726 + 0.0112i)7-s + (0.994 + 0.103i)8-s + (0.0844 + 0.0974i)9-s + (0.741 − 0.798i)10-s + (−1.46 − 0.211i)11-s + (0.933 + 0.507i)12-s + (0.507 + 0.326i)13-s + (0.00686 + 0.0115i)14-s + (1.05 − 0.480i)15-s + (0.990 + 0.137i)16-s + (−1.58 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0680i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 - 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.29556 + 0.146251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.29556 + 0.146251i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.99 - 0.138i)T \) |
| 23 | \( 1 + (-164. - 502. i)T \) |
| good | 3 | \( 1 + (-8.69 - 3.97i)T + (53.0 + 61.2i)T^{2} \) |
| 5 | \( 1 + (-17.8 + 20.5i)T + (-88.9 - 618. i)T^{2} \) |
| 7 | \( 1 + (-0.355 - 0.553i)T + (-997. + 2.18e3i)T^{2} \) |
| 11 | \( 1 + (177. + 25.5i)T + (1.40e4 + 4.12e3i)T^{2} \) |
| 13 | \( 1 + (-85.7 - 55.1i)T + (1.18e4 + 2.59e4i)T^{2} \) |
| 17 | \( 1 + (456. + 134. i)T + (7.02e4 + 4.51e4i)T^{2} \) |
| 19 | \( 1 + (-173. - 589. i)T + (-1.09e5 + 7.04e4i)T^{2} \) |
| 29 | \( 1 + (144. + 42.3i)T + (5.95e5 + 3.82e5i)T^{2} \) |
| 31 | \( 1 + (-449. + 205. i)T + (6.04e5 - 6.97e5i)T^{2} \) |
| 37 | \( 1 + (900. + 1.03e3i)T + (-2.66e5 + 1.85e6i)T^{2} \) |
| 41 | \( 1 + (808. - 932. i)T + (-4.02e5 - 2.79e6i)T^{2} \) |
| 43 | \( 1 + (-1.10e3 - 503. i)T + (2.23e6 + 2.58e6i)T^{2} \) |
| 47 | \( 1 + 1.00e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (3.69e3 - 2.37e3i)T + (3.27e6 - 7.17e6i)T^{2} \) |
| 59 | \( 1 + (-725. + 1.12e3i)T + (-5.03e6 - 1.10e7i)T^{2} \) |
| 61 | \( 1 + (2.14e3 + 4.69e3i)T + (-9.06e6 + 1.04e7i)T^{2} \) |
| 67 | \( 1 + (1.58e3 - 227. i)T + (1.93e7 - 5.67e6i)T^{2} \) |
| 71 | \( 1 + (857. - 123. i)T + (2.43e7 - 7.15e6i)T^{2} \) |
| 73 | \( 1 + (-6.65e3 + 1.95e3i)T + (2.38e7 - 1.53e7i)T^{2} \) |
| 79 | \( 1 + (332. - 517. i)T + (-1.61e7 - 3.54e7i)T^{2} \) |
| 83 | \( 1 + (2.57e3 - 2.23e3i)T + (6.75e6 - 4.69e7i)T^{2} \) |
| 89 | \( 1 + (-1.20e3 + 2.63e3i)T + (-4.10e7 - 4.74e7i)T^{2} \) |
| 97 | \( 1 + (-1.10e4 + 1.27e4i)T + (-1.25e7 - 8.76e7i)T^{2} \) |
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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
−13.53738163410663495725749591568, −12.72209916785133126217846131409, −11.32048124869804925526286367228, −10.04058462237434379295681145150, −8.892877108880268319591964071856, −7.76297120468937085104178310966, −5.98777455425558137369328026963, −4.88933428845562489452579610872, −3.44559414153867363617666277400, −2.03137855594724266257479114034,
2.28353683106756818119164975877, 2.91469617388098370068207139616, 4.90582828863019440561811266351, 6.39534013320070101402051914372, 7.39771027556581992907590196036, 8.687803387464118140345922202841, 10.43116497828769715037061241275, 11.07371698184505990018336531257, 12.86227427166415178435640508141, 13.43619686640785301130509027967
