L(s) = 1 | + 3-s − 5-s + 0.549i·7-s + 9-s − 1.98i·11-s − 2.38i·13-s − 15-s − 4.49·17-s + (−4.14 + 1.35i)19-s + 0.549i·21-s − 2.88i·23-s + 25-s + 27-s + 3.54i·29-s + 0.0281·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.207i·7-s + 0.333·9-s − 0.598i·11-s − 0.662i·13-s − 0.258·15-s − 1.09·17-s + (−0.950 + 0.309i)19-s + 0.119i·21-s − 0.602i·23-s + 0.200·25-s + 0.192·27-s + 0.658i·29-s + 0.00505·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5788805788\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5788805788\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (4.14 - 1.35i)T \) |
good | 7 | \( 1 - 0.549iT - 7T^{2} \) |
| 11 | \( 1 + 1.98iT - 11T^{2} \) |
| 13 | \( 1 + 2.38iT - 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 + 2.88iT - 23T^{2} \) |
| 29 | \( 1 - 3.54iT - 29T^{2} \) |
| 31 | \( 1 - 0.0281T + 31T^{2} \) |
| 37 | \( 1 + 3.37iT - 37T^{2} \) |
| 41 | \( 1 - 4.40iT - 41T^{2} \) |
| 43 | \( 1 - 6.83iT - 43T^{2} \) |
| 47 | \( 1 - 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 3.79T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 7.95iT - 83T^{2} \) |
| 89 | \( 1 + 0.0323iT - 89T^{2} \) |
| 97 | \( 1 - 0.596iT - 97T^{2} \) |
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
−8.745309844930971811526272161849, −7.913159395211633241980874935754, −7.34445428615974462763728141716, −6.37276688778603609968535352013, −5.84152852768893272209529869280, −4.64177702996062602301861536585, −4.17204698186785707073239197493, −3.09842441499079502478294393530, −2.53739950022576723538010145523, −1.28723816146179000988413952599,
0.14237984690237156538186712112, 1.76031649134197587338873866781, 2.42261232832788478686943091094, 3.57893495439844407489530399044, 4.24331137083147362457439237933, 4.81540033445201857559474869108, 5.94584999527185613860983449925, 6.93812829295592863589190212138, 7.15710279980435331858686679926, 8.138135961181193451734442689206