| L(s) = 1 | + (−0.00473 − 1.41i)2-s + (−0.494 + 1.19i)3-s + (−1.99 + 0.0133i)4-s + (−1.78 + 3.49i)5-s + (1.68 + 0.693i)6-s + (−0.375 − 1.56i)7-s + (0.0283 + 2.82i)8-s + (0.942 + 0.942i)9-s + (4.94 + 2.50i)10-s + (−2.53 + 2.96i)11-s + (0.972 − 2.39i)12-s + (1.25 + 2.05i)13-s + (−2.21 + 0.539i)14-s + (−3.28 − 3.84i)15-s + (3.99 − 0.0535i)16-s + (−0.283 − 3.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.00334 − 0.999i)2-s + (−0.285 + 0.688i)3-s + (−0.999 + 0.00669i)4-s + (−0.796 + 1.56i)5-s + (0.689 + 0.282i)6-s + (−0.142 − 0.591i)7-s + (0.0100 + 0.999i)8-s + (0.314 + 0.314i)9-s + (1.56 + 0.790i)10-s + (−0.762 + 0.893i)11-s + (0.280 − 0.690i)12-s + (0.348 + 0.568i)13-s + (−0.591 + 0.144i)14-s + (−0.848 − 0.993i)15-s + (0.999 − 0.0133i)16-s + (−0.0688 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562997 + 0.391846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562997 + 0.391846i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.00473 + 1.41i)T \) |
| 41 | \( 1 + (-3.29 - 5.48i)T \) |
| good | 3 | \( 1 + (0.494 - 1.19i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (1.78 - 3.49i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.375 + 1.56i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (2.53 - 2.96i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 2.05i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.283 + 3.60i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (3.76 + 2.30i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-7.12 - 5.17i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.227 + 2.88i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-0.405 + 1.24i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.54 - 7.82i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.293 + 1.85i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (1.22 - 5.11i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-6.09 - 0.479i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-0.993 + 1.36i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 6.77i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-2.97 + 2.53i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-3.90 - 3.33i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (4.47 - 4.47i)T - 73iT^{2} \) |
| 79 | \( 1 + (-10.4 - 4.32i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 5.70iT - 83T^{2} \) |
| 89 | \( 1 + (3.03 - 0.728i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.39 + 2.90i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.00809770593483465292781132888, −11.50657817550872019129013170533, −11.07122163969765522747930532663, −10.29286196216418057201795153570, −9.529699697549291025820979211282, −7.79602731896521258155468918252, −6.88752986799622561842025902393, −4.86520482228311124305953312162, −3.92359745413742539352670336797, −2.65514540010961937610317457952,
0.70510621221458293260172336314, 3.94328602112492213738842869330, 5.26077494805171273868321983228, 6.13898299038000962082967352245, 7.52604128417264414150394287079, 8.518138279713066688889244394759, 8.907670732988921582458888142194, 10.64306290683515264473574221129, 12.22418292919373925433415790547, 12.81708516176586033750386886537
