L(s) = 1 | + (1 + i)2-s − 1.73i·3-s + 2i·4-s + (−0.866 + 1.5i)5-s + (1.73 − 1.73i)6-s + (−1.73 + 2i)7-s + (−2 + 2i)8-s − 2.99·9-s + (−2.36 + 0.633i)10-s + (2.5 + 4.33i)11-s + 3.46·12-s + (−0.866 − 1.5i)13-s + (−3.73 + 0.267i)14-s + (2.59 + 1.49i)15-s − 4·16-s + (1.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s − 0.999i·3-s + i·4-s + (−0.387 + 0.670i)5-s + (0.707 − 0.707i)6-s + (−0.654 + 0.755i)7-s + (−0.707 + 0.707i)8-s − 0.999·9-s + (−0.748 + 0.200i)10-s + (0.753 + 1.30i)11-s + 1.00·12-s + (−0.240 − 0.416i)13-s + (−0.997 + 0.0716i)14-s + (0.670 + 0.387i)15-s − 16-s + (0.363 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.724043 + 1.28940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.724043 + 1.28940i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 5 | \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.866 + 1.5i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.06 - 3.5i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.5 + 6.06i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + (-9.52 - 5.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (-13.5 - 7.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.80148186856340096776050181752, −10.40597652513523957253361902331, −9.164655508807905146282543133654, −8.172261251449355891314383585271, −7.38090039808502827257865916177, −6.54299872048024963407176290608, −6.04266861940329737826507298603, −4.66201839809633003629275355393, −3.30530371173192085313716048738, −2.31304894017500441067379821647,
0.70012247849513531544941636128, 2.91737771817043246453735946834, 3.95574820829292894217575661107, 4.44977000388254875768371618797, 5.72991353229310829738351881843, 6.53310558806318440545704429545, 8.231944208967156654316478293042, 9.127691906393613551824657549623, 9.952524423308190468903987825497, 10.58469735495602496522593579631