L(s) = 1 | + (1.29 + 0.565i)2-s + (0.284 − 1.70i)3-s + (1.35 + 1.46i)4-s + (0.5 − 0.866i)5-s + (1.33 − 2.05i)6-s + (2.24 − 1.29i)7-s + (0.933 + 2.67i)8-s + (−2.83 − 0.970i)9-s + (1.13 − 0.839i)10-s + (−3.78 + 2.18i)11-s + (2.89 − 1.90i)12-s + (2.94 + 1.69i)13-s + (3.64 − 0.410i)14-s + (−1.33 − 1.10i)15-s + (−0.301 + 3.98i)16-s − 8.12i·17-s + ⋯ |
L(s) = 1 | + (0.916 + 0.400i)2-s + (0.163 − 0.986i)3-s + (0.679 + 0.733i)4-s + (0.223 − 0.387i)5-s + (0.544 − 0.838i)6-s + (0.849 − 0.490i)7-s + (0.329 + 0.944i)8-s + (−0.946 − 0.323i)9-s + (0.359 − 0.265i)10-s + (−1.14 + 0.658i)11-s + (0.834 − 0.550i)12-s + (0.816 + 0.471i)13-s + (0.974 − 0.109i)14-s + (−0.345 − 0.284i)15-s + (−0.0753 + 0.997i)16-s − 1.96i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.45385 - 0.424152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45385 - 0.424152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 - 0.565i)T \) |
| 3 | \( 1 + (-0.284 + 1.70i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-2.24 + 1.29i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.78 - 2.18i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 - 1.69i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 8.12iT - 17T^{2} \) |
| 19 | \( 1 - 0.0767T + 19T^{2} \) |
| 23 | \( 1 + (1.81 - 3.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.63 - 8.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.23 + 3.60i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.03iT - 37T^{2} \) |
| 41 | \( 1 + (-0.999 - 0.577i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.09 - 5.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.608 + 1.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + (12.8 + 7.40i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 6.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0137 + 0.0237i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.518T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 + (-0.0609 + 0.0351i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.39 - 4.84i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.30iT - 89T^{2} \) |
| 97 | \( 1 + (-0.938 - 1.62i)T + (-48.5 + 84.0i)T^{2} \) |
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
−11.56079099063700048762064324453, −10.96464795193550951476895011982, −9.335229278497837436594958195613, −8.118501137671730911013003956611, −7.53075666192689563244736663891, −6.68400826241882597964753175994, −5.41056369271325633803445091556, −4.67955036919774223242168915081, −3.02975922491266520990742727737, −1.70366636584354550720549565033,
2.18193811693162326498729564259, 3.35688089571332915819652315975, 4.40006552316423054169072119359, 5.60487998212823463435249422204, 6.04005362854710777349620575988, 7.941920051455992142261420884483, 8.684621978091862848650833783972, 10.16573349720918131124122056267, 10.70924415251972207082639770398, 11.20385313798461037183877108745