| L(s) = 1 | + (−0.651 − 0.228i)2-s + (0.820 + 2.66i)3-s + (−1.19 − 0.949i)4-s + (0.777 − 1.47i)5-s + (0.0718 − 1.92i)6-s + (−2.29 − 1.32i)7-s + (1.29 + 2.05i)8-s + (−3.93 + 2.67i)9-s + (−0.842 + 0.781i)10-s + (4.19 − 0.792i)11-s + (1.55 − 3.94i)12-s + (−3.48 − 0.907i)13-s + (1.19 + 1.38i)14-s + (4.55 + 0.861i)15-s + (0.304 + 1.33i)16-s + (5.05 + 6.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.460 − 0.161i)2-s + (0.473 + 1.53i)3-s + (−0.595 − 0.474i)4-s + (0.347 − 0.658i)5-s + (0.0293 − 0.784i)6-s + (−0.865 − 0.500i)7-s + (0.457 + 0.728i)8-s + (−1.31 + 0.893i)9-s + (−0.266 + 0.247i)10-s + (1.26 − 0.239i)11-s + (0.447 − 1.14i)12-s + (−0.967 − 0.251i)13-s + (0.318 + 0.370i)14-s + (1.17 + 0.222i)15-s + (0.0760 + 0.333i)16-s + (1.22 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.900174 + 0.668632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.900174 + 0.668632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.29 + 1.32i)T \) |
| 13 | \( 1 + (3.48 + 0.907i)T \) |
| good | 2 | \( 1 + (0.651 + 0.228i)T + (1.56 + 1.24i)T^{2} \) |
| 3 | \( 1 + (-0.820 - 2.66i)T + (-2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.777 + 1.47i)T + (-2.81 - 4.13i)T^{2} \) |
| 11 | \( 1 + (-4.19 + 0.792i)T + (10.2 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.05 - 6.33i)T + (-3.78 + 16.5i)T^{2} \) |
| 19 | \( 1 + (-1.45 - 5.43i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-6.51 - 5.19i)T + (5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (6.63 - 0.999i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (1.21 + 4.52i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.49 + 0.281i)T + (36.0 + 8.23i)T^{2} \) |
| 41 | \( 1 + (-6.47 + 0.242i)T + (40.8 - 3.06i)T^{2} \) |
| 43 | \( 1 + (3.77 + 4.07i)T + (-3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.0208 - 0.110i)T + (-43.7 + 17.1i)T^{2} \) |
| 53 | \( 1 + (-2.75 - 7.01i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-4.53 + 7.22i)T + (-25.5 - 53.1i)T^{2} \) |
| 61 | \( 1 + (0.776 + 5.15i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (1.49 - 5.56i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.44 - 3.31i)T + (-20.9 + 67.8i)T^{2} \) |
| 73 | \( 1 + (2.42 + 0.458i)T + (67.9 + 26.6i)T^{2} \) |
| 79 | \( 1 + (-3.90 - 6.77i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.15 - 1.80i)T + (64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (1.67 + 4.79i)T + (-69.5 + 55.4i)T^{2} \) |
| 97 | \( 1 + (-13.9 - 3.73i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.31537743183111820295913688025, −9.693543745197209055403406748662, −9.377483629855911968166569010911, −8.647437419038485724637042327054, −7.53614296352471796114480800647, −5.82355688942765788693493727120, −5.24327647878854125696182259451, −3.99039334031755774687002917455, −3.50400631401614744938354391890, −1.37202909620918688212622198425,
0.78615038709183435322757272320, 2.53578432047523289368494849129, 3.23404631023182740741456670467, 4.96638206731086087018672890741, 6.46978579616039040398711912608, 7.09816237931916928843190950631, 7.41988020627255390319528931277, 8.853643365013388219588436675689, 9.203397994062402248290812146075, 10.01335370482421762349838195893
