L(s) = 1 | + (1.12 − 0.767i)2-s + (−0.0507 + 0.0471i)3-s + (−0.0524 + 0.133i)4-s + (0.822 + 0.253i)5-s + (−0.0210 + 0.0920i)6-s + (−2.32 − 1.26i)7-s + (0.649 + 2.84i)8-s + (−0.223 + 2.98i)9-s + (1.12 − 0.345i)10-s + (0.329 + 4.40i)11-s + (−0.00363 − 0.00925i)12-s + (−0.900 − 0.433i)13-s + (−3.58 + 0.359i)14-s + (−0.0537 + 0.0258i)15-s + (2.70 + 2.51i)16-s + (7.62 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (0.796 − 0.542i)2-s + (−0.0293 + 0.0272i)3-s + (−0.0262 + 0.0668i)4-s + (0.367 + 0.113i)5-s + (−0.00857 + 0.0375i)6-s + (−0.878 − 0.478i)7-s + (0.229 + 1.00i)8-s + (−0.0746 + 0.995i)9-s + (0.354 − 0.109i)10-s + (0.0994 + 1.32i)11-s + (−0.00104 − 0.00267i)12-s + (−0.249 − 0.120i)13-s + (−0.958 + 0.0961i)14-s + (−0.0138 + 0.00667i)15-s + (0.676 + 0.627i)16-s + (1.84 + 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88943 + 0.703483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88943 + 0.703483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.32 + 1.26i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
good | 2 | \( 1 + (-1.12 + 0.767i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (0.0507 - 0.0471i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.822 - 0.253i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.329 - 4.40i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (-7.62 - 1.14i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.00360 - 0.00624i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.66 + 0.551i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (3.01 - 3.78i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.04 + 5.19i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.13 - 9.35i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.997 + 4.36i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-7.00 + 4.77i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.126 + 0.321i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (2.64 - 0.816i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.59 - 6.60i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (8.07 + 13.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.71 - 9.67i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (5.85 + 3.99i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-8.47 + 14.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.51 - 2.17i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.0121 - 0.162i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56724342689875681603706152417, −10.12482450918693002129790189954, −9.155324009607595940380274631615, −7.77290469554851339461520022665, −7.30211110568127551961094536712, −5.86612999460171494822971076469, −5.02885273872231149665323264519, −4.03818192316051840822569968999, −3.03082291904189090390861132655, −1.92724486686308693235981084901,
0.904654687015108496038747794512, 3.12146452443859104627717894017, 3.78394357183939532349347963962, 5.40372625176899883127132085662, 5.80130257884190137431116962637, 6.55815688150904821603587115664, 7.57723216603821898978708013516, 8.980532249135521497524210539016, 9.516752360801296917152685769776, 10.29163280759817937136342085111