| L(s) = 1 | + (−1.07 + 2.03i)2-s + (−0.725 + 0.687i)3-s + (−1.84 − 2.72i)4-s + (−3.23 + 0.712i)5-s + (−0.615 − 2.21i)6-s + (1.31 − 0.996i)7-s + (2.94 − 0.320i)8-s + (0.0541 − 0.998i)9-s + (2.03 − 7.34i)10-s + (−0.922 − 2.31i)11-s + (3.21 + 0.707i)12-s + (0.119 + 2.21i)13-s + (0.612 + 3.73i)14-s + (1.85 − 2.74i)15-s + (−0.0892 + 0.224i)16-s + (−2.34 − 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.761 + 1.43i)2-s + (−0.419 + 0.397i)3-s + (−0.922 − 1.36i)4-s + (−1.44 + 0.318i)5-s + (−0.251 − 0.904i)6-s + (0.495 − 0.376i)7-s + (1.04 − 0.113i)8-s + (0.0180 − 0.332i)9-s + (0.644 − 2.32i)10-s + (−0.278 − 0.698i)11-s + (0.927 + 0.204i)12-s + (0.0332 + 0.613i)13-s + (0.163 + 0.998i)14-s + (0.479 − 0.707i)15-s + (−0.0223 + 0.0560i)16-s + (−0.567 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0262805 - 0.0208608i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0262805 - 0.0208608i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.725 - 0.687i)T \) |
| 59 | \( 1 + (-0.287 - 7.67i)T \) |
| good | 2 | \( 1 + (1.07 - 2.03i)T + (-1.12 - 1.65i)T^{2} \) |
| 5 | \( 1 + (3.23 - 0.712i)T + (4.53 - 2.09i)T^{2} \) |
| 7 | \( 1 + (-1.31 + 0.996i)T + (1.87 - 6.74i)T^{2} \) |
| 11 | \( 1 + (0.922 + 2.31i)T + (-7.98 + 7.56i)T^{2} \) |
| 13 | \( 1 + (-0.119 - 2.21i)T + (-12.9 + 1.40i)T^{2} \) |
| 17 | \( 1 + (2.34 + 1.77i)T + (4.54 + 16.3i)T^{2} \) |
| 19 | \( 1 + (6.93 - 2.33i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (-2.37 + 1.42i)T + (10.7 - 20.3i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 2.15i)T + (-16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (7.57 + 2.55i)T + (24.6 + 18.7i)T^{2} \) |
| 37 | \( 1 + (9.97 + 1.08i)T + (36.1 + 7.95i)T^{2} \) |
| 41 | \( 1 + (9.34 + 5.62i)T + (19.2 + 36.2i)T^{2} \) |
| 43 | \( 1 + (2.62 - 6.58i)T + (-31.2 - 29.5i)T^{2} \) |
| 47 | \( 1 + (-6.67 - 1.47i)T + (42.6 + 19.7i)T^{2} \) |
| 53 | \( 1 + (-0.271 - 0.979i)T + (-45.4 + 27.3i)T^{2} \) |
| 61 | \( 1 + (-1.09 + 2.07i)T + (-34.2 - 50.4i)T^{2} \) |
| 67 | \( 1 + (-16.2 + 1.76i)T + (65.4 - 14.4i)T^{2} \) |
| 71 | \( 1 + (6.36 + 1.40i)T + (64.4 + 29.8i)T^{2} \) |
| 73 | \( 1 + (1.45 + 8.86i)T + (-69.1 + 23.3i)T^{2} \) |
| 79 | \( 1 + (-9.22 - 8.73i)T + (4.27 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-0.0827 + 0.0974i)T + (-13.4 - 81.9i)T^{2} \) |
| 89 | \( 1 + (-3.29 - 6.21i)T + (-49.9 + 73.6i)T^{2} \) |
| 97 | \( 1 + (2.41 - 14.7i)T + (-91.9 - 30.9i)T^{2} \) |
| 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
−13.93367029847444315491272064863, −12.31095778915651135423567790842, −11.17484750636462875258172930330, −10.56002719864930760380216760809, −8.979186710614588522641063995417, −8.291900165014785774184190204637, −7.29917938629328513902669954362, −6.49980123728282974863773738648, −5.06626567373164487420415586940, −3.87428672739843394173520386364,
0.04034576621935261007288414586, 1.99999807620495900989029805301, 3.69963290678108894168760506292, 4.95802320535889488589781806693, 7.01242116069878487375601227318, 8.266175120008075031105924849221, 8.724096529252406805517539830420, 10.31382888985548554202715617406, 11.05395813870996983219312130329, 11.78070067895392743909963372427
