| L(s) = 1 | + (−1.59 − 2.35i)2-s + (−0.162 + 2.99i)3-s + (−0.0415 + 0.104i)4-s + (14.5 − 6.71i)5-s + (7.32 − 4.40i)6-s + (0.937 − 3.37i)7-s + (−21.9 + 4.82i)8-s + (−8.94 − 0.973i)9-s + (−39.0 − 23.4i)10-s + (10.9 − 10.3i)11-s + (−0.305 − 0.141i)12-s + (83.7 − 9.10i)13-s + (−9.45 + 3.18i)14-s + (17.7 + 44.5i)15-s + (47.1 + 44.6i)16-s + (−3.89 − 14.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.565 − 0.833i)2-s + (−0.0312 + 0.576i)3-s + (−0.00519 + 0.0130i)4-s + (1.29 − 0.600i)5-s + (0.498 − 0.299i)6-s + (0.0506 − 0.182i)7-s + (−0.969 + 0.213i)8-s + (−0.331 − 0.0360i)9-s + (−1.23 − 0.742i)10-s + (0.299 − 0.283i)11-s + (−0.00734 − 0.00340i)12-s + (1.78 − 0.194i)13-s + (−0.180 + 0.0608i)14-s + (0.305 + 0.767i)15-s + (0.736 + 0.697i)16-s + (−0.0555 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.995756 - 1.21975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995756 - 1.21975i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.162 - 2.99i)T \) |
| 59 | \( 1 + (-441. + 103. i)T \) |
| good | 2 | \( 1 + (1.59 + 2.35i)T + (-2.96 + 7.43i)T^{2} \) |
| 5 | \( 1 + (-14.5 + 6.71i)T + (80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-0.937 + 3.37i)T + (-293. - 176. i)T^{2} \) |
| 11 | \( 1 + (-10.9 + 10.3i)T + (72.0 - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-83.7 + 9.10i)T + (2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (3.89 + 14.0i)T + (-4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (83.9 - 63.8i)T + (1.83e3 - 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-61.5 + 116. i)T + (-6.82e3 - 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-141. + 208. i)T + (-9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (220. + 167. i)T + (7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (114. + 25.2i)T + (4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (100. + 190. i)T + (-3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (179. + 169. i)T + (4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (-301. - 139. i)T + (6.72e4 + 7.91e4i)T^{2} \) |
| 53 | \( 1 + (12.7 - 7.65i)T + (6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-441. - 650. i)T + (-8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (441. - 97.2i)T + (2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (128. + 59.5i)T + (2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (650. - 219. i)T + (3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-38.7 - 715. i)T + (-4.90e5 + 5.33e4i)T^{2} \) |
| 83 | \( 1 + (101. + 620. i)T + (-5.41e5 + 1.82e5i)T^{2} \) |
| 89 | \( 1 + (83.2 - 122. i)T + (-2.60e5 - 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-889. - 299. i)T + (7.26e5 + 5.52e5i)T^{2} \) |
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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
−11.65827400722683900344499996581, −10.61641107951889290726387110882, −10.18419921092614462885348232119, −8.981577678189456040510341217097, −8.615958261453472845558331847112, −6.25356311044486617208430157779, −5.63784731456415107132594289354, −3.92888697256736334269498181883, −2.23929833948666973410066590324, −0.932593774249935737316802152891,
1.64169213609628775798654983267, 3.23451505500721230429972044921, 5.57331591982861689657397169110, 6.51762024091222541419569522376, 7.00976171260171896693299717814, 8.573866238057958693777826003980, 9.056897931558253242288693714568, 10.43121112131070573583987515969, 11.42086488706944884214371021428, 12.75026066791170413836041549061
