| L(s) = 1 | + 11.7i·5-s + 53.2·7-s + 124. i·11-s − 74.6·13-s + 7.70i·17-s + 54.1·19-s − 399. i·23-s + 486.·25-s + 540. i·29-s + 1.53e3·31-s + 627. i·35-s − 1.71e3·37-s + 1.25e3i·41-s + 2.60e3·43-s − 800. i·47-s + ⋯ |
| L(s) = 1 | + 0.471i·5-s + 1.08·7-s + 1.03i·11-s − 0.441·13-s + 0.0266i·17-s + 0.149·19-s − 0.756i·23-s + 0.777·25-s + 0.642i·29-s + 1.59·31-s + 0.512i·35-s − 1.25·37-s + 0.749i·41-s + 1.41·43-s − 0.362i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.342919937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342919937\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 11.7iT - 625T^{2} \) |
| 7 | \( 1 - 53.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 124. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 74.6T + 2.85e4T^{2} \) |
| 17 | \( 1 - 7.70iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 54.1T + 1.30e5T^{2} \) |
| 23 | \( 1 + 399. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 540. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.53e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.71e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.25e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.60e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 800. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.33e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 15.0T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.18e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.92e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 949.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 237.T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.31e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 575. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.51e4T + 8.85e7T^{2} \) |
| 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
−9.256979593654561003032695766585, −8.480993763972760476646317094186, −7.57795336946395800906972560773, −7.02240864407560447773465126024, −6.03971068762959332941779573789, −4.83188623922463622698675349298, −4.48618923443973612501139920325, −3.03901251843940475758832853205, −2.13212818577923400884185404776, −1.08076036736390509993263442631,
0.51226952419249827361436490443, 1.44021390928757119958994332191, 2.60860625150699030442978122356, 3.75690782835403294379042801723, 4.80403538047130576839158153460, 5.37665595958360212338390511400, 6.36951516277166737883424748341, 7.43586795103745088266855876985, 8.206164421144511680704959724699, 8.737256806895729106917464269414
