| L(s) = 1 | + (−1.21 + 3.59i)2-s + (−2.84 + 0.941i)3-s + (−8.26 − 6.28i)4-s + (1.37 + 0.547i)5-s + (0.0639 − 11.3i)6-s + (8.82 − 4.08i)7-s + (20.0 − 13.5i)8-s + (7.22 − 5.36i)9-s + (−3.63 + 4.27i)10-s + (−18.6 − 5.19i)11-s + (29.4 + 10.1i)12-s + (3.26 − 6.16i)13-s + (3.98 + 36.6i)14-s + (−4.43 − 0.265i)15-s + (13.4 + 48.4i)16-s + (2.10 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.605 + 1.79i)2-s + (−0.949 + 0.313i)3-s + (−2.06 − 1.57i)4-s + (0.275 + 0.109i)5-s + (0.0106 − 1.89i)6-s + (1.26 − 0.583i)7-s + (2.50 − 1.69i)8-s + (0.802 − 0.596i)9-s + (−0.363 + 0.427i)10-s + (−1.69 − 0.471i)11-s + (2.45 + 0.842i)12-s + (0.251 − 0.474i)13-s + (0.284 + 2.61i)14-s + (−0.295 − 0.0176i)15-s + (0.840 + 3.02i)16-s + (0.123 − 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.614083 + 0.288539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614083 + 0.288539i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.84 - 0.941i)T \) |
| 59 | \( 1 + (-15.5 + 56.9i)T \) |
| good | 2 | \( 1 + (1.21 - 3.59i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-1.37 - 0.547i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-8.82 + 4.08i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (18.6 + 5.19i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-3.26 + 6.16i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-2.10 + 4.54i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-1.65 + 0.364i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-27.8 - 4.55i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (8.47 + 25.1i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (-27.9 - 6.15i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-25.3 + 37.3i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-60.8 + 9.97i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-6.18 - 22.2i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (52.5 - 20.9i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-5.17 + 4.39i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (11.5 + 3.89i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (67.8 + 100. i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (52.6 - 20.9i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-101. + 11.0i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-62.9 + 37.8i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-5.99 + 0.325i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (12.5 + 37.1i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-23.2 - 2.52i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−13.00337180727367912671258237451, −11.11720413330975745370187827419, −10.48539751629710257051557837138, −9.475100385856492269934592874568, −8.024037012169993017988084830926, −7.58138932281650602608917687821, −6.19631892428004826444753403255, −5.33306617643074390652023108558, −4.59088120709789227370028968198, −0.64995823572762557986083437801,
1.35604660219239026400755551657, 2.49366259479570482684937690898, 4.57834000166449648269981694853, 5.36296706864519497998662104853, 7.57965687091395383276822456156, 8.460488899373433220820549999058, 9.706375725986911463454082937345, 10.67543069238611468725110787448, 11.23990005217443148249055085727, 12.02992575966812460560873419013
