L(s) = 1 | + 1.73i·3-s + 3.46·5-s − 2i·7-s − 2.99·9-s + 3.46i·11-s + 5.99i·15-s + 3.46·21-s + 6.99·25-s − 5.19i·27-s − 10.3·29-s − 10i·31-s − 5.99·33-s − 6.92i·35-s − 10.3·45-s + 3·49-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 1.54·5-s − 0.755i·7-s − 0.999·9-s + 1.04i·11-s + 1.54i·15-s + 0.755·21-s + 1.39·25-s − 0.999i·27-s − 1.92·29-s − 1.79i·31-s − 1.04·33-s − 1.17i·35-s − 1.54·45-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29581 + 0.536744i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29581 + 0.536744i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 - 17.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−12.90649354745569504346093811782, −11.42080890996111399603065516516, −10.40680032789476037594704772750, −9.761821418677728907157702231770, −9.135013810216059280878062108629, −7.53245001390505133599562759371, −6.16299848642427505324257100487, −5.17668557154051799532013434223, −3.96613534954855246776242677496, −2.18965243836546082585507017635,
1.68969402465132796719893245960, 2.93265545998427546509763475109, 5.50558532915946811707201965240, 5.94156164681195485811015626370, 7.10870659391611060195736760098, 8.574031530437562596343786013299, 9.191494158685320040968530216438, 10.50037454426579202494958929812, 11.55688822363957266631092405266, 12.61866298150081840950699032976