| L(s) = 1 | + (−0.283 + 0.246i)3-s + (−2.45 + 0.352i)5-s + (1.80 − 3.94i)7-s + (−0.406 + 2.82i)9-s + (−0.456 + 1.55i)11-s + (−1.85 + 0.845i)13-s + (0.609 − 0.703i)15-s + (−1.97 − 1.26i)17-s + (−1.17 − 1.82i)19-s + (0.459 + 1.56i)21-s + (−4.47 − 1.71i)23-s + (1.09 − 0.322i)25-s + (−1.19 − 1.85i)27-s + (−2.85 + 4.43i)29-s + (−3.79 + 4.37i)31-s + ⋯ |
| L(s) = 1 | + (−0.163 + 0.142i)3-s + (−1.09 + 0.157i)5-s + (0.681 − 1.49i)7-s + (−0.135 + 0.943i)9-s + (−0.137 + 0.468i)11-s + (−0.513 + 0.234i)13-s + (0.157 − 0.181i)15-s + (−0.478 − 0.307i)17-s + (−0.268 − 0.417i)19-s + (0.100 + 0.341i)21-s + (−0.933 − 0.358i)23-s + (0.219 − 0.0645i)25-s + (−0.229 − 0.356i)27-s + (−0.529 + 0.824i)29-s + (−0.680 + 0.785i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29607\times10^{-6} - 0.00103179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29607\times10^{-6} - 0.00103179i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (4.47 + 1.71i)T \) |
| good | 3 | \( 1 + (0.283 - 0.246i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (2.45 - 0.352i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 3.94i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.456 - 1.55i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.85 - 0.845i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.97 + 1.26i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.17 + 1.82i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.85 - 4.43i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.79 - 4.37i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.95 + 0.856i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0399 - 0.277i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.34 + 2.89i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + (11.4 + 5.23i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (10.7 - 4.91i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 8.70i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (4.29 + 14.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (2.37 - 0.697i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.91 + 5.73i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.23 + 2.70i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.47 + 0.212i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.94 - 6.86i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 11.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.75656256979291806389397932042, −10.33675329801980800492301756607, −9.034764215019800429419666044541, −7.87265393254578334927814729413, −7.53512773525566205384543440546, −6.73975554653513618841339363110, −5.04532992319034109751285725998, −4.49061820685797632744843760239, −3.58593900731368435084136835357, −1.93321739525066670353643647137,
0.00051493848211003505510498687, 2.01626496115370923919866211979, 3.37304275096060085648806537224, 4.39538498165122808515081174670, 5.58464058040877494017798381521, 6.21612543559431059959708290353, 7.58411925786843276562465958050, 8.227865192649236274531466857340, 8.936223296131847940576303923248, 9.817960607287275720872506423016
