| L(s) = 1 | + (0.654 − 0.755i)3-s + (2.24 − 1.44i)5-s + (2.21 + 0.886i)7-s + (−0.142 − 0.989i)9-s + (0.240 + 5.05i)11-s + (3.25 − 0.310i)13-s + (0.380 − 2.64i)15-s + (0.620 + 2.55i)17-s + (−0.689 + 0.276i)19-s + (2.11 − 1.09i)21-s + (0.936 + 0.180i)23-s + (0.894 − 1.95i)25-s + (−0.841 − 0.540i)27-s + (0.974 − 1.68i)29-s + (−7.90 − 0.754i)31-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.436i)3-s + (1.00 − 0.646i)5-s + (0.836 + 0.334i)7-s + (−0.0474 − 0.329i)9-s + (0.0726 + 1.52i)11-s + (0.902 − 0.0861i)13-s + (0.0982 − 0.683i)15-s + (0.150 + 0.619i)17-s + (−0.158 + 0.0633i)19-s + (0.462 − 0.238i)21-s + (0.195 + 0.0376i)23-s + (0.178 − 0.391i)25-s + (−0.161 − 0.104i)27-s + (0.180 − 0.313i)29-s + (−1.41 − 0.135i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28442 - 0.392878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28442 - 0.392878i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-2.69 + 7.73i)T \) |
| good | 5 | \( 1 + (-2.24 + 1.44i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (-2.21 - 0.886i)T + (5.06 + 4.83i)T^{2} \) |
| 11 | \( 1 + (-0.240 - 5.05i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (-3.25 + 0.310i)T + (12.7 - 2.46i)T^{2} \) |
| 17 | \( 1 + (-0.620 - 2.55i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (0.689 - 0.276i)T + (13.7 - 13.1i)T^{2} \) |
| 23 | \( 1 + (-0.936 - 0.180i)T + (21.3 + 8.54i)T^{2} \) |
| 29 | \( 1 + (-0.974 + 1.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.90 + 0.754i)T + (30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (3.51 + 6.08i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.77 - 1.69i)T + (1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-2.03 - 0.597i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (4.00 + 11.5i)T + (-36.9 + 29.0i)T^{2} \) |
| 53 | \( 1 + (4.25 - 1.25i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-5.12 - 11.2i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.457 + 9.60i)T + (-60.7 - 5.79i)T^{2} \) |
| 71 | \( 1 + (2.64 - 10.8i)T + (-63.1 - 32.5i)T^{2} \) |
| 73 | \( 1 + (-0.635 + 13.3i)T + (-72.6 - 6.93i)T^{2} \) |
| 79 | \( 1 + (-3.59 - 5.04i)T + (-25.8 + 74.6i)T^{2} \) |
| 83 | \( 1 + (1.66 + 0.855i)T + (48.1 + 67.6i)T^{2} \) |
| 89 | \( 1 + (-2.92 - 3.37i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.507 - 0.878i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.06205643266600199892464105652, −9.253364669132920220341015778876, −8.601411423279025451616423324630, −7.74365459275347502122430896627, −6.76808147449735612743825675240, −5.72218076850023691770199193105, −4.97900902125137321779601059668, −3.81028935948818221545905742747, −2.07620885663629578299841838495, −1.58337493700762003657991704384,
1.44158380985333904021144792127, 2.81884127879933928320014950635, 3.71230906321792484763512690705, 5.02375015822836118511990514999, 5.88522736485392665137256335497, 6.73225823997616791003447330261, 7.913775415440677786943541118678, 8.683448046324028210915631275350, 9.423379797439360088446119608892, 10.44121232329674806273382268620
