| L(s) = 1 | + (0.173 + 0.984i)3-s + (−1.59 + 1.33i)5-s + (−1.26 + 2.19i)7-s + (−0.939 + 0.342i)9-s + (−2.13 − 3.68i)11-s + (−0.349 + 1.98i)13-s + (−1.59 − 1.33i)15-s + (−1.63 − 0.593i)17-s + (−4.14 − 1.34i)19-s + (−2.37 − 0.866i)21-s + (−0.961 − 0.806i)23-s + (−0.115 + 0.654i)25-s + (−0.5 − 0.866i)27-s + (1.51 − 0.552i)29-s + (−4.39 + 7.61i)31-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.568i)3-s + (−0.713 + 0.598i)5-s + (−0.478 + 0.828i)7-s + (−0.313 + 0.114i)9-s + (−0.642 − 1.11i)11-s + (−0.0968 + 0.549i)13-s + (−0.411 − 0.345i)15-s + (−0.395 − 0.143i)17-s + (−0.951 − 0.308i)19-s + (−0.519 − 0.188i)21-s + (−0.200 − 0.168i)23-s + (−0.0230 + 0.130i)25-s + (−0.0962 − 0.166i)27-s + (0.281 − 0.102i)29-s + (−0.789 + 1.36i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0507894 + 0.568953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0507894 + 0.568953i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (4.14 + 1.34i)T \) |
| good | 5 | \( 1 + (1.59 - 1.33i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.26 - 2.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.13 + 3.68i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.349 - 1.98i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.63 + 0.593i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.961 + 0.806i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.51 + 0.552i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.39 - 7.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 + (-0.930 - 5.27i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.53 - 4.64i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.85 + 0.673i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.87 - 2.41i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (14.0 + 5.10i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.22 - 2.70i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 4.04i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (1.48 - 1.24i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.364 - 2.06i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.01 - 5.74i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.600 + 1.04i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.96 - 16.8i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (5.30 + 1.93i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.19425248788275228370332178937, −10.86061945407315476572643849587, −9.650654521223632551580495087805, −8.789568012295128134011720212281, −8.046943572838188265687062926865, −6.78836013783524867566465851629, −5.89562950663144713787478105222, −4.69733906890954080385343311873, −3.47486438425349992240831084526, −2.60209887255179244034515601433,
0.33029769129077782845096074644, 2.19222055985013099747987345477, 3.79549360347662942650134240624, 4.67802566099692791964647119584, 6.03376082542959211173804996151, 7.19497582478313160829192583227, 7.76918242986327805026677031197, 8.655854883457552919119698762361, 9.844006146183182615874919962717, 10.59394326907745812653809312998
