L(s) = 1 | + (0.5 − 0.866i)2-s + (0.991 − 1.41i)3-s + (−0.499 − 0.866i)4-s + (−3.72 + 2.14i)5-s + (−0.733 − 1.56i)6-s + (−1.08 + 2.41i)7-s − 0.999·8-s + (−1.03 − 2.81i)9-s + 4.29i·10-s − 4.52·11-s + (−1.72 − 0.148i)12-s + (2.01 + 2.98i)13-s + (1.54 + 2.14i)14-s + (−0.639 + 7.41i)15-s + (−0.5 + 0.866i)16-s + (−0.0962 − 0.166i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.572 − 0.819i)3-s + (−0.249 − 0.433i)4-s + (−1.66 + 0.960i)5-s + (−0.299 − 0.640i)6-s + (−0.410 + 0.911i)7-s − 0.353·8-s + (−0.344 − 0.938i)9-s + 1.35i·10-s − 1.36·11-s + (−0.498 − 0.0429i)12-s + (0.559 + 0.828i)13-s + (0.413 + 0.573i)14-s + (−0.165 + 1.91i)15-s + (−0.125 + 0.216i)16-s + (−0.0233 − 0.0404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.187434 + 0.229001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.187434 + 0.229001i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.991 + 1.41i)T \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
| 13 | \( 1 + (-2.01 - 2.98i)T \) |
good | 5 | \( 1 + (3.72 - 2.14i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 17 | \( 1 + (0.0962 + 0.166i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + (-1.22 - 0.704i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (8.66 - 5.00i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.16 - 2.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.93 + 2.85i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.24 - 0.717i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.73 + 2.99i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.70 + 5.02i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.59 + 3.22i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.17 + 1.83i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 8.11iT - 61T^{2} \) |
| 67 | \( 1 - 5.05iT - 67T^{2} \) |
| 71 | \( 1 + (5.59 - 9.68i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.90 + 10.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.93 + 3.34i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.15iT - 83T^{2} \) |
| 89 | \( 1 + (1.87 + 1.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.86 - 11.8i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.22688743323035160300006960885, −10.51083153857260975522750553153, −9.083838108673492908099129630970, −8.401721475198996155987220729784, −7.42065422268188055755925660148, −6.73563008008179406109716607335, −5.48984262332437463990076249849, −3.90396615948474617066975919828, −3.16241226601926959039813056244, −2.22888694088001808974006805319,
0.13493807144763938057931723416, 3.16604347864180635088170201366, 3.95123777579058416155618258309, 4.67550668789487109023193170939, 5.62230731170722265507459574252, 7.38391964427461397287395612041, 7.88287991417385487757857537307, 8.473775862918435690451302183264, 9.494472583191072326056998038058, 10.66972340319889636819833390292