L(s) = 1 | + (1.09 + 0.894i)2-s + 1.88·3-s + (0.400 + 1.95i)4-s − 2.41·5-s + (2.06 + 1.68i)6-s − 2.57i·7-s + (−1.31 + 2.50i)8-s + 0.564·9-s + (−2.64 − 2.15i)10-s − 0.736·11-s + (0.756 + 3.69i)12-s − 4.07i·13-s + (2.29 − 2.81i)14-s − 4.55·15-s + (−3.67 + 1.56i)16-s + (3.99 + 1.02i)17-s + ⋯ |
L(s) = 1 | + (0.774 + 0.632i)2-s + 1.09·3-s + (0.200 + 0.979i)4-s − 1.07·5-s + (0.844 + 0.689i)6-s − 0.971i·7-s + (−0.464 + 0.885i)8-s + 0.188·9-s + (−0.836 − 0.682i)10-s − 0.221·11-s + (0.218 + 1.06i)12-s − 1.13i·13-s + (0.614 − 0.752i)14-s − 1.17·15-s + (−0.919 + 0.392i)16-s + (0.968 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63725 + 0.726666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63725 + 0.726666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.09 - 0.894i)T \) |
| 17 | \( 1 + (-3.99 - 1.02i)T \) |
good | 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 + 2.57iT - 7T^{2} \) |
| 11 | \( 1 + 0.736T + 11T^{2} \) |
| 13 | \( 1 + 4.07iT - 13T^{2} \) |
| 19 | \( 1 + 0.853iT - 19T^{2} \) |
| 23 | \( 1 - 8.20iT - 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 - 3.51iT - 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 11.7iT - 41T^{2} \) |
| 43 | \( 1 + 1.09iT - 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 4.66iT - 53T^{2} \) |
| 59 | \( 1 - 12.5iT - 59T^{2} \) |
| 61 | \( 1 - 8.16T + 61T^{2} \) |
| 67 | \( 1 + 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 0.511iT - 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.45iT - 79T^{2} \) |
| 83 | \( 1 - 4.35iT - 83T^{2} \) |
| 89 | \( 1 - 5.72T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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
−13.67080963465926241941032890372, −12.61614242147944457725058712956, −11.60005636487600525787787245443, −10.31005193158426428881403210930, −8.664436524364988235427299414834, −7.78736402041580649452835917125, −7.29979723330198269967493990461, −5.44243141608483339860734891371, −3.87255242277334385840362261062, −3.18438639728596498808018829283,
2.40443976251949198853723594244, 3.53037113445320946768690074860, 4.78988517901047572031799521909, 6.41174707648920794650941228677, 7.974050639428045852476235947516, 8.889666669288098501932456488753, 9.993814996767281962003860679602, 11.48965035561529620843933862503, 12.01429293318177120736943794642, 13.03207994726407152519957797920