L(s) = 1 | + (−0.834 + 0.699i)3-s + (3.00 + 1.09i)5-s + (0.278 − 0.482i)7-s + (−0.315 + 1.78i)9-s + (−1.96 − 3.39i)11-s + (−3.19 − 2.67i)13-s + (−3.27 + 1.19i)15-s + (−0.660 − 3.74i)17-s + (−1.84 + 3.94i)19-s + (0.105 + 0.597i)21-s + (4.67 − 1.70i)23-s + (4.01 + 3.36i)25-s + (−2.62 − 4.53i)27-s + (0.0201 − 0.114i)29-s + (−3.54 + 6.13i)31-s + ⋯ |
L(s) = 1 | + (−0.481 + 0.404i)3-s + (1.34 + 0.489i)5-s + (0.105 − 0.182i)7-s + (−0.105 + 0.595i)9-s + (−0.591 − 1.02i)11-s + (−0.885 − 0.743i)13-s + (−0.845 + 0.307i)15-s + (−0.160 − 0.907i)17-s + (−0.423 + 0.906i)19-s + (0.0229 + 0.130i)21-s + (0.974 − 0.354i)23-s + (0.803 + 0.673i)25-s + (−0.504 − 0.873i)27-s + (0.00374 − 0.0212i)29-s + (−0.636 + 1.10i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.908797 + 0.208693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.908797 + 0.208693i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.84 - 3.94i)T \) |
good | 3 | \( 1 + (0.834 - 0.699i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.00 - 1.09i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.278 + 0.482i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.96 + 3.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.19 + 2.67i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.660 + 3.74i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-4.67 + 1.70i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0201 + 0.114i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.54 - 6.13i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 + (-9.20 + 7.72i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.74 - 2.45i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.00419 - 0.0237i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (8.18 - 2.97i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.88 - 10.7i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (11.7 - 4.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.27 + 12.8i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.17 - 1.15i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.338 + 0.284i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.31 + 1.10i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.90 - 5.02i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.83 - 1.53i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 10.5i)T + (-91.1 + 33.1i)T^{2} \) |
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
−14.34061974395676903792073792611, −13.72003487473720876579840719793, −12.52201386551504989283397396602, −10.76387517127708070766473489228, −10.52489651494538019896668980920, −9.174095304755864053013076302092, −7.55392246888851403107939255643, −5.94784578336753004732095096395, −5.09272918690391610210629142956, −2.67993320624799818480210094506,
2.05046261599182544128095747837, 4.83062161754843451185583484967, 6.02769541008150040947497117186, 7.18938297363131295390613072542, 9.051653752364956815312864369710, 9.796543658751649818778285741020, 11.21612627929494277208417656494, 12.60615347896638500951284125645, 13.01449916753309294567366676325, 14.42132497675847109177316524668