L(s) = 1 | + (−1.27 + 0.620i)2-s + (−0.331 + 0.573i)3-s + (1.23 − 1.57i)4-s + (−0.866 + 0.5i)5-s + (0.0650 − 0.934i)6-s + (1.68 + 2.03i)7-s + (−0.585 + 2.76i)8-s + (1.28 + 2.21i)9-s + (0.790 − 1.17i)10-s + (−3.12 − 1.80i)11-s + (0.496 + 1.22i)12-s + 5.83i·13-s + (−3.40 − 1.54i)14-s − 0.662i·15-s + (−0.971 − 3.88i)16-s + (−1.18 − 0.684i)17-s + ⋯ |
L(s) = 1 | + (−0.898 + 0.438i)2-s + (−0.191 + 0.331i)3-s + (0.615 − 0.788i)4-s + (−0.387 + 0.223i)5-s + (0.0265 − 0.381i)6-s + (0.637 + 0.770i)7-s + (−0.207 + 0.978i)8-s + (0.426 + 0.739i)9-s + (0.249 − 0.370i)10-s + (−0.943 − 0.544i)11-s + (0.143 + 0.354i)12-s + 1.61i·13-s + (−0.911 − 0.412i)14-s − 0.171i·15-s + (−0.242 − 0.970i)16-s + (−0.287 − 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.429729 + 0.502457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.429729 + 0.502457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.27 - 0.620i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-1.68 - 2.03i)T \) |
good | 3 | \( 1 + (0.331 - 0.573i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (3.12 + 1.80i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.83iT - 13T^{2} \) |
| 17 | \( 1 + (1.18 + 0.684i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 - 3.54i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.81 + 1.62i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + (-4.43 + 7.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.39 + 9.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.832iT - 41T^{2} \) |
| 43 | \( 1 + 3.10iT - 43T^{2} \) |
| 47 | \( 1 + (-3.44 - 5.97i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.70 + 6.42i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.73 - 6.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.28 + 0.742i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.19 + 1.26i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.52iT - 71T^{2} \) |
| 73 | \( 1 + (4.47 + 2.58i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.82 + 5.67i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.49T + 83T^{2} \) |
| 89 | \( 1 + (-8.13 + 4.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.343iT - 97T^{2} \) |
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
−13.72129936115081069986972597263, −12.00318013990465754771492973335, −11.18911309388176204591734568704, −10.38981687988725798409625443039, −9.171110431016083641940245214174, −8.202900965750597166751472683145, −7.26146889021893056004942462102, −5.84398153348070646060525552190, −4.66882494005313933162087658763, −2.22467448367355201241763399500,
0.979798289880133223118971040087, 3.15412406017387629948178650013, 4.86586028865057141587915270550, 6.84376475237165648654058418335, 7.67626344895164899478811944619, 8.563895097832375659844534326054, 10.04096036903949313605887241067, 10.66572398350288176189847199196, 11.81553753186252686640181186241, 12.66258161134400617275120154437