| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 33·5-s − 36·6-s + 49·7-s + 64·8-s + 81·9-s + 132·10-s − 94·11-s − 144·12-s − 169·13-s + 196·14-s − 297·15-s + 256·16-s − 1.11e3·17-s + 324·18-s − 1.90e3·19-s + 528·20-s − 441·21-s − 376·22-s + 961·23-s − 576·24-s − 2.03e3·25-s − 676·26-s − 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.590·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.417·10-s − 0.234·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.340·15-s + 1/4·16-s − 0.933·17-s + 0.235·18-s − 1.21·19-s + 0.295·20-s − 0.218·21-s − 0.165·22-s + 0.378·23-s − 0.204·24-s − 0.651·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 33 T + p^{5} T^{2} \) |
| 11 | \( 1 + 94 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1112 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1909 T + p^{5} T^{2} \) |
| 23 | \( 1 - 961 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6337 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7015 T + p^{5} T^{2} \) |
| 37 | \( 1 - 928 T + p^{5} T^{2} \) |
| 41 | \( 1 - 11442 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12711 T + p^{5} T^{2} \) |
| 47 | \( 1 + 15107 T + p^{5} T^{2} \) |
| 53 | \( 1 - 18691 T + p^{5} T^{2} \) |
| 59 | \( 1 + 12360 T + p^{5} T^{2} \) |
| 61 | \( 1 - 14110 T + p^{5} T^{2} \) |
| 67 | \( 1 + 53746 T + p^{5} T^{2} \) |
| 71 | \( 1 + 47748 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25301 T + p^{5} T^{2} \) |
| 79 | \( 1 + 5447 T + p^{5} T^{2} \) |
| 83 | \( 1 - 29393 T + p^{5} T^{2} \) |
| 89 | \( 1 + 77621 T + p^{5} T^{2} \) |
| 97 | \( 1 - 73607 T + p^{5} T^{2} \) |
| 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
−9.791986166736818930708542501070, −8.680324806852511866283287919743, −7.55946809212350755846172983022, −6.55229469626350225188951769932, −5.84540840289370594528512432424, −4.88635843304958650860313594006, −4.09290257510662530825531539691, −2.56124328774597524177449317981, −1.61913246233736735192058645368, 0,
1.61913246233736735192058645368, 2.56124328774597524177449317981, 4.09290257510662530825531539691, 4.88635843304958650860313594006, 5.84540840289370594528512432424, 6.55229469626350225188951769932, 7.55946809212350755846172983022, 8.680324806852511866283287919743, 9.791986166736818930708542501070
