L(s) = 1 | + i·3-s − 2.33·5-s + (0.490 + 2.59i)7-s − 9-s + 0.304·11-s − 5.46·13-s − 2.33i·15-s − 6.37i·17-s + 0.840i·19-s + (−2.59 + 0.490i)21-s − 0.111i·23-s + 0.449·25-s − i·27-s − 5.46i·29-s − 7.64·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.04·5-s + (0.185 + 0.982i)7-s − 0.333·9-s + 0.0918·11-s − 1.51·13-s − 0.602i·15-s − 1.54i·17-s + 0.192i·19-s + (−0.567 + 0.107i)21-s − 0.0232i·23-s + 0.0899·25-s − 0.192i·27-s − 1.01i·29-s − 1.37·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0353849 - 0.171125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0353849 - 0.171125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-0.490 - 2.59i)T \) |
good | 5 | \( 1 + 2.33T + 5T^{2} \) |
| 11 | \( 1 - 0.304T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 6.37iT - 17T^{2} \) |
| 19 | \( 1 - 0.840iT - 19T^{2} \) |
| 23 | \( 1 + 0.111iT - 23T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 - 6.44iT - 37T^{2} \) |
| 41 | \( 1 - 8.66iT - 41T^{2} \) |
| 43 | \( 1 + 6.06T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 - 8.35T + 67T^{2} \) |
| 71 | \( 1 - 2.07iT - 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 7.90iT - 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.08iT - 89T^{2} \) |
| 97 | \( 1 - 6.42iT - 97T^{2} \) |
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
−11.29950939174111866333760415867, −9.842476375079634952635773034875, −9.481473027034595794624290302214, −8.325618327511666047033336121193, −7.67512047906875517465942746053, −6.63819951454249203682169984231, −5.23636458728775407242791254170, −4.73024317275051043071020037798, −3.45151029770352820443677560902, −2.40215731217090857015507591652,
0.088563085662121899552612416098, 1.81752742441727891184984816143, 3.44719253469941768885172865616, 4.28584431747901619517847317103, 5.41950882301081536777131466829, 6.80033205796769354473542311955, 7.41172566083761350283892773047, 8.013647624507454483266258952165, 9.011861916348681118027426132563, 10.21084619941188819870262240399