| L(s) = 1 | + (−0.194 + 0.648i)2-s + (1.28 + 0.847i)4-s + (0.373 − 0.866i)5-s + (0.0849 + 1.45i)7-s + (−1.83 + 1.54i)8-s + (0.488 + 0.410i)10-s + (3.65 − 4.90i)11-s + (3.60 − 3.81i)13-s + (−0.961 − 0.227i)14-s + (0.579 + 1.34i)16-s + (−0.00536 + 0.0304i)17-s + (0.634 + 3.60i)19-s + (1.21 − 0.799i)20-s + (2.47 + 3.32i)22-s + (−0.00607 + 0.104i)23-s + ⋯ |
| L(s) = 1 | + (−0.137 + 0.458i)2-s + (0.644 + 0.423i)4-s + (0.167 − 0.387i)5-s + (0.0321 + 0.551i)7-s + (−0.649 + 0.544i)8-s + (0.154 + 0.129i)10-s + (1.10 − 1.48i)11-s + (0.998 − 1.05i)13-s + (−0.257 − 0.0609i)14-s + (0.144 + 0.335i)16-s + (−0.00130 + 0.00737i)17-s + (0.145 + 0.826i)19-s + (0.271 − 0.178i)20-s + (0.527 + 0.708i)22-s + (−0.00126 + 0.0217i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74159 + 0.645163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74159 + 0.645163i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.194 - 0.648i)T + (-1.67 - 1.09i)T^{2} \) |
| 5 | \( 1 + (-0.373 + 0.866i)T + (-3.43 - 3.63i)T^{2} \) |
| 7 | \( 1 + (-0.0849 - 1.45i)T + (-6.95 + 0.812i)T^{2} \) |
| 11 | \( 1 + (-3.65 + 4.90i)T + (-3.15 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-3.60 + 3.81i)T + (-0.755 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.00536 - 0.0304i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.634 - 3.60i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.00607 - 0.104i)T + (-22.8 - 2.67i)T^{2} \) |
| 29 | \( 1 + (-0.498 + 0.118i)T + (25.9 - 13.0i)T^{2} \) |
| 31 | \( 1 + (7.30 - 3.66i)T + (18.5 - 24.8i)T^{2} \) |
| 37 | \( 1 + (-5.42 + 1.97i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (1.25 + 4.20i)T + (-34.2 + 22.5i)T^{2} \) |
| 43 | \( 1 + (1.36 + 0.159i)T + (41.8 + 9.91i)T^{2} \) |
| 47 | \( 1 + (-5.01 - 2.51i)T + (28.0 + 37.6i)T^{2} \) |
| 53 | \( 1 + (4.89 - 8.47i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.37 + 4.53i)T + (-16.9 + 56.5i)T^{2} \) |
| 61 | \( 1 + (1.43 - 0.943i)T + (24.1 - 56.0i)T^{2} \) |
| 67 | \( 1 + (-2.66 - 0.631i)T + (59.8 + 30.0i)T^{2} \) |
| 71 | \( 1 + (-1.66 - 1.39i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (6.38 - 5.35i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.61 + 12.0i)T + (-66.0 - 43.4i)T^{2} \) |
| 83 | \( 1 + (5.11 - 17.0i)T + (-69.3 - 45.6i)T^{2} \) |
| 89 | \( 1 + (10.6 - 8.96i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.77 + 6.43i)T + (-66.5 + 70.5i)T^{2} \) |
| 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
−10.78435784950427976437975290735, −9.264186015251582700000667534959, −8.655555629639035415656339898519, −8.062694679986703401668144890290, −6.96398629068374194750279078811, −5.81697044576245218518377196825, −5.70806849284112965498812243948, −3.77704732607129201347406453854, −2.99835582083569696398035471939, −1.34089487692540863413732642236,
1.31570522949528101330926002988, 2.30070570813307827733603569846, 3.70584027210713563429773982850, 4.63386410445898473546182682566, 6.16386422914054621528556472985, 6.77666869667211288964017487471, 7.38294920013099233306194653449, 8.938837381395693726711036982240, 9.568667961011290164954504501183, 10.33367610724957715004942756975
