L(s) = 1 | + (−0.258 + 0.965i)2-s + (−1.34 − 1.08i)3-s + (−0.866 − 0.499i)4-s + (0.429 + 0.429i)5-s + (1.39 − 1.02i)6-s + (−0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (0.644 + 2.92i)9-s + (−0.526 + 0.303i)10-s + (1.44 + 0.388i)11-s + (0.626 + 1.61i)12-s + (−1.82 − 3.11i)13-s − i·14-s + (−0.113 − 1.04i)15-s + (0.500 + 0.866i)16-s + (2.50 − 4.33i)17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.779 − 0.626i)3-s + (−0.433 − 0.249i)4-s + (0.192 + 0.192i)5-s + (0.570 − 0.417i)6-s + (−0.365 + 0.0978i)7-s + (0.249 − 0.249i)8-s + (0.214 + 0.976i)9-s + (−0.166 + 0.0960i)10-s + (0.436 + 0.117i)11-s + (0.180 + 0.466i)12-s + (−0.505 − 0.862i)13-s − 0.267i·14-s + (−0.0293 − 0.270i)15-s + (0.125 + 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.939161 - 0.0813870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939161 - 0.0813870i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (1.34 + 1.08i)T \) |
| 7 | \( 1 + (0.965 - 0.258i)T \) |
| 13 | \( 1 + (1.82 + 3.11i)T \) |
good | 5 | \( 1 + (-0.429 - 0.429i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.44 - 0.388i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.50 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.744 - 2.77i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.87 - 4.97i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.53 + 3.19i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.86 + 2.86i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.29 + 8.56i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.973 + 3.63i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-9.35 - 5.39i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.10 + 7.10i)T - 47iT^{2} \) |
| 53 | \( 1 - 7.88iT - 53T^{2} \) |
| 59 | \( 1 + (-3.35 - 12.5i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (0.879 - 1.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.7 + 2.87i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (5.54 - 1.48i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (10.8 + 10.8i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 + (5.15 + 5.15i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.57 - 1.76i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.785 + 2.92i)T + (-84.0 + 48.5i)T^{2} \) |
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
−10.60640043679228490038873384091, −9.952444789602710355268368878157, −8.984895387952033887051119361020, −7.62450678186545025249671703348, −7.33306289807387594275381896188, −6.07755967755655501363251988860, −5.62284504540613614622840262667, −4.42045534372986115323919117237, −2.68056582485744999975943568044, −0.812687340131946737170521902249,
1.15999222581680790061280880051, 2.99362884486956079261139225789, 4.18572346557555800784368140016, 4.98015254758219127224128306202, 6.18943880524840956303834522407, 7.04335375925348065974508675225, 8.571816997590962262773917206467, 9.302420985244657778577855388823, 10.05896998277911436438284273783, 10.77592217093493060944955457126