L(s) = 1 | + (−0.610 − 0.416i)2-s + (−2.18 − 2.02i)3-s + (−0.531 − 1.35i)4-s + (−2.47 + 0.764i)5-s + (0.489 + 2.14i)6-s + (1.83 − 1.90i)7-s + (−0.567 + 2.48i)8-s + (0.439 + 5.86i)9-s + (1.83 + 0.564i)10-s + (−0.116 + 1.55i)11-s + (−1.58 + 4.03i)12-s + (−0.900 + 0.433i)13-s + (−1.91 + 0.399i)14-s + (6.96 + 3.35i)15-s + (−0.750 + 0.696i)16-s + (−5.91 + 0.892i)17-s + ⋯ |
L(s) = 1 | + (−0.431 − 0.294i)2-s + (−1.26 − 1.17i)3-s + (−0.265 − 0.676i)4-s + (−1.10 + 0.342i)5-s + (0.199 + 0.875i)6-s + (0.693 − 0.720i)7-s + (−0.200 + 0.879i)8-s + (0.146 + 1.95i)9-s + (0.579 + 0.178i)10-s + (−0.0351 + 0.469i)11-s + (−0.457 + 1.16i)12-s + (−0.249 + 0.120i)13-s + (−0.511 + 0.106i)14-s + (1.79 + 0.866i)15-s + (−0.187 + 0.174i)16-s + (−1.43 + 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.284217 - 0.0144414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.284217 - 0.0144414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.83 + 1.90i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
good | 2 | \( 1 + (0.610 + 0.416i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (2.18 + 2.02i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.47 - 0.764i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.116 - 1.55i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (5.91 - 0.892i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.151 + 0.262i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.12 - 0.470i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.28 - 2.86i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.43 + 5.94i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 3.90i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.928 - 4.06i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.421 - 1.84i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.522i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (3.77 + 9.61i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.99 - 0.925i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (3.37 - 8.60i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (6.59 - 11.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.25 - 2.82i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-8.48 + 5.78i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 3.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.4 + 6.45i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.22 - 16.3i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−10.93506278706659000034213421655, −10.04066376771965148082936616031, −8.662403877257688401489338566170, −7.73144165044480710844746976770, −7.06666877982096091786813362292, −6.26531365795708384314280339520, −5.03505256940072239906515132105, −4.33021789570435413525926074800, −2.14136572490024046820988770695, −0.912232594685121175851847521847,
0.29389964541160533993253525758, 3.20441125165342176221320386480, 4.49836752247923246274566214485, 4.69012164221285238663671991704, 6.01413899457229614517272332966, 7.07005952406028789666604587877, 8.246085209928017027543783460088, 8.769307786861768908106084427650, 9.626504019431692548173547631885, 10.79637017290868360926879180554