L(s) = 1 | + (−0.154 + 0.227i)2-s + (−0.819 + 1.52i)3-s + (0.712 + 1.78i)4-s + (−1.09 + 2.37i)5-s + (−0.221 − 0.422i)6-s + (−0.787 − 2.83i)7-s + (−1.05 − 0.232i)8-s + (−1.65 − 2.50i)9-s + (−0.370 − 0.616i)10-s + (−1.07 − 1.02i)11-s + (−3.31 − 0.378i)12-s + (−0.324 + 2.98i)13-s + (0.767 + 0.258i)14-s + (−2.71 − 3.61i)15-s + (−2.57 + 2.44i)16-s + (6.60 + 1.83i)17-s + ⋯ |
L(s) = 1 | + (−0.109 + 0.161i)2-s + (−0.473 + 0.880i)3-s + (0.356 + 0.893i)4-s + (−0.490 + 1.06i)5-s + (−0.0902 − 0.172i)6-s + (−0.297 − 1.07i)7-s + (−0.372 − 0.0820i)8-s + (−0.552 − 0.833i)9-s + (−0.117 − 0.194i)10-s + (−0.325 − 0.308i)11-s + (−0.955 − 0.109i)12-s + (−0.0899 + 0.826i)13-s + (0.205 + 0.0691i)14-s + (−0.702 − 0.934i)15-s + (−0.644 + 0.610i)16-s + (1.60 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.256317 + 0.771378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.256317 + 0.771378i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.819 - 1.52i)T \) |
| 59 | \( 1 + (-6.44 + 4.17i)T \) |
good | 2 | \( 1 + (0.154 - 0.227i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (1.09 - 2.37i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.787 + 2.83i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (1.07 + 1.02i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.324 - 2.98i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (-6.60 - 1.83i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 1.45i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-3.50 - 6.61i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (6.12 - 4.15i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (4.49 + 5.91i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (0.431 + 1.96i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (2.48 + 1.31i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (-4.78 - 5.05i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-7.93 + 3.67i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (1.20 - 2.00i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-10.2 - 6.98i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 5.95i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (0.638 + 1.37i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (0.995 - 2.95i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.685 + 12.6i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (-1.16 + 7.10i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (5.81 + 8.57i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-0.296 - 0.879i)T + (-77.2 + 58.7i)T^{2} \) |
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
−12.98364325599633501394207490453, −11.73697928217625901470093115740, −11.14843009592630128931587452506, −10.29242975266089525036535927153, −9.209251482155195652511271758901, −7.60477181089819166180963117910, −7.10937662708509962157391963027, −5.71787245026561326679284581636, −3.80731487675416615590681325163, −3.37553745014991375201966291925,
0.840525129204220449961659375973, 2.58919159521125800567061840646, 5.22047497449214034170508393691, 5.58838624788083738297407931836, 7.04631757210011198463978674563, 8.202420482987091163192699764484, 9.236879688105339053595407070932, 10.42123898264496532964590208983, 11.55440798607293227657508262430, 12.38661597777260028137795897983