L(s) = 1 | + (1.90 + 3.16i)2-s + (−2.63 + 1.42i)3-s + (−4.50 + 8.50i)4-s + (0.853 + 7.84i)5-s + (−9.53 − 5.62i)6-s + (12.1 − 4.09i)7-s + (−20.7 + 1.12i)8-s + (4.92 − 7.53i)9-s + (−23.1 + 17.6i)10-s + (−2.40 − 1.62i)11-s + (−0.247 − 28.8i)12-s + (−1.30 − 1.23i)13-s + (36.0 + 30.6i)14-s + (−13.4 − 19.4i)15-s + (−21.4 − 31.6i)16-s + (5.27 − 15.6i)17-s + ⋯ |
L(s) = 1 | + (0.951 + 1.58i)2-s + (−0.879 + 0.475i)3-s + (−1.12 + 2.12i)4-s + (0.170 + 1.56i)5-s + (−1.58 − 0.938i)6-s + (1.73 − 0.584i)7-s + (−2.59 + 0.140i)8-s + (0.546 − 0.837i)9-s + (−2.31 + 1.76i)10-s + (−0.218 − 0.148i)11-s + (−0.0206 − 2.40i)12-s + (−0.100 − 0.0949i)13-s + (2.57 + 2.18i)14-s + (−0.896 − 1.29i)15-s + (−1.33 − 1.97i)16-s + (0.310 − 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.371502 - 1.93759i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.371502 - 1.93759i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.63 - 1.42i)T \) |
| 59 | \( 1 + (-12.2 - 57.7i)T \) |
good | 2 | \( 1 + (-1.90 - 3.16i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.853 - 7.84i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (-12.1 + 4.09i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (2.40 + 1.62i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (1.30 + 1.23i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-5.27 + 15.6i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (0.0242 + 0.148i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (-30.3 + 8.42i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (1.00 - 1.66i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (6.01 - 36.6i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (1.35 - 24.9i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (64.5 + 17.9i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (19.8 + 29.2i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-4.02 + 37.0i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-18.5 + 24.4i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-39.1 + 23.5i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (2.79 + 51.5i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-2.48 + 22.8i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (20.7 - 24.3i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (23.3 - 58.6i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-9.41 - 20.3i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (22.7 - 37.8i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-0.456 - 0.537i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−13.53148919226304952353081396622, −12.00132926324323937273502078474, −11.14967585311126743450331717315, −10.29497184736777534055910094116, −8.493625766069237575534388107939, −7.16967175412834329417328503724, −6.88925633543185483344380753521, −5.42393702098264772935350351939, −4.79281361028479700514321234555, −3.41324721099811065175115559742,
1.15149636546473640097814206336, 1.94802160223955241161650084957, 4.40560055749878163601531839778, 5.08178340970432186914459928715, 5.69415135837553781439226537224, 8.015802688179970617316979672698, 9.116288064104726259940746115737, 10.40926131737383266510158431964, 11.45510200433307397761660416624, 11.81501079060920962026951979916