L(s) = 1 | + (0.988 − 1.01i)2-s + (−0.0442 − 1.99i)4-s + 1.12i·5-s + (−2.06 − 1.93i)8-s + (1.14 + 1.11i)10-s + 1.35i·11-s + 5.58i·13-s + (−3.99 + 0.177i)16-s + 3.97i·17-s + 6.14·19-s + (2.25 − 0.0500i)20-s + (1.36 + 1.33i)22-s + 6.61i·23-s + 3.72·25-s + (5.64 + 5.52i)26-s + ⋯ |
L(s) = 1 | + (0.699 − 0.714i)2-s + (−0.0221 − 0.999i)4-s + 0.504i·5-s + (−0.730 − 0.683i)8-s + (0.360 + 0.353i)10-s + 0.407i·11-s + 1.54i·13-s + (−0.999 + 0.0442i)16-s + 0.963i·17-s + 1.40·19-s + (0.504 − 0.0111i)20-s + (0.290 + 0.284i)22-s + 1.37i·23-s + 0.745·25-s + (1.10 + 1.08i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.262154798\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.262154798\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.988 + 1.01i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 11 | \( 1 - 1.35iT - 11T^{2} \) |
| 13 | \( 1 - 5.58iT - 13T^{2} \) |
| 17 | \( 1 - 3.97iT - 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 6.61iT - 23T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 0.149iT - 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 0.733T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 9.11iT - 61T^{2} \) |
| 67 | \( 1 - 0.234iT - 67T^{2} \) |
| 71 | \( 1 + 8.74iT - 71T^{2} \) |
| 73 | \( 1 - 1.76iT - 73T^{2} \) |
| 79 | \( 1 - 8.40iT - 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 7.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.23iT - 97T^{2} \) |
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
−9.472208057287002699100102514282, −8.883186047671831346699212895277, −7.41740976378390683169847439056, −6.94192093622769688717309109901, −5.88901713474394511821802716697, −5.22600884758472897730616225913, −4.07292054362374173647955351397, −3.54596506373356432775792288875, −2.31341819715209911230591892701, −1.43517920561097840851495720472,
0.70123594313566234270088409998, 2.67521262362641180170451339780, 3.37622099366267173595870610116, 4.52666990481256582459033631194, 5.32550399670735450351902808922, 5.80238557959193709756873015022, 6.91048841715987466961583600637, 7.66529752161647686645730123201, 8.265009574830218944974394887135, 9.105272105110329495048764746042