L(s) = 1 | + (−1.25 − 2.17i)2-s + (−2.16 + 3.74i)4-s + (−0.257 − 0.445i)7-s + 5.83·8-s + (−1.66 − 2.87i)11-s + (−0.660 + 1.14i)13-s + (−0.646 + 1.11i)14-s + (−3.01 − 5.22i)16-s − 3.32·17-s − 1.32·19-s + (−4.17 + 7.23i)22-s + (−2.06 + 3.57i)23-s + 3.32·26-s + 2.22·28-s + (−0.693 − 1.20i)29-s + ⋯ |
L(s) = 1 | + (−0.888 − 1.53i)2-s + (−1.08 + 1.87i)4-s + (−0.0971 − 0.168i)7-s + 2.06·8-s + (−0.500 − 0.867i)11-s + (−0.183 + 0.317i)13-s + (−0.172 + 0.299i)14-s + (−0.753 − 1.30i)16-s − 0.805·17-s − 0.303·19-s + (−0.890 + 1.54i)22-s + (−0.430 + 0.745i)23-s + 0.651·26-s + 0.419·28-s + (−0.128 − 0.222i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151302 + 0.0762082i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151302 + 0.0762082i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (0.257 + 0.445i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.660 - 1.14i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 + (2.06 - 3.57i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.693 + 1.20i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.36 - 7.56i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.292T + 37T^{2} \) |
| 41 | \( 1 + (5.67 - 9.82i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.17 - 8.96i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.43 - 4.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 + (2.51 - 4.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.72 + 8.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.99T + 71T^{2} \) |
| 73 | \( 1 + 6.05T + 73T^{2} \) |
| 79 | \( 1 + (-4.02 - 6.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.771 + 1.33i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (6.12 + 10.6i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74651688250928569446287698472, −9.858532044225941022271358248443, −9.111864215430295811881893306792, −8.376433017271665840814665942565, −7.53903692349321517755555575038, −6.23518883236906577892360513863, −4.78177851836079994314696131305, −3.64500975778967013080768340106, −2.73715037349610513840168530819, −1.50283543159909682813111091049,
0.12071564774306242017003622109, 2.19740894355872632037123980973, 4.22945125352899184850075259714, 5.27707201120011134649945293660, 6.07516870060887400959711536572, 7.07339385492898213723954687590, 7.59920587065523431602482155485, 8.604511452173781805202304358472, 9.190951332440742806523464121799, 10.13814898538878533740929239386