| L(s) = 1 | − 19·13-s + 107·19-s − 125·25-s − 289·31-s + 323·37-s + 71·43-s + 182·61-s − 127·67-s − 271·73-s − 1.38e3·79-s − 1.33e3·97-s − 1.80e3·103-s + 2.21e3·109-s + ⋯ |
| L(s) = 1 | − 0.405·13-s + 1.29·19-s − 25-s − 1.67·31-s + 1.43·37-s + 0.251·43-s + 0.382·61-s − 0.231·67-s − 0.434·73-s − 1.97·79-s − 1.39·97-s − 1.72·103-s + 1.94·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 289 T + p^{3} T^{2} \) |
| 37 | \( 1 - 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 71 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 + 127 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 271 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
| 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
−8.545843733685224632714726384741, −7.61390269976383885758839905506, −7.16288905228318448190789636938, −5.99066425361125673322565969009, −5.38094305432879129202478260369, −4.36414772300130078373076281064, −3.45137754740827241561592182715, −2.43687262764253220589504777569, −1.29628243589113686695676520543, 0,
1.29628243589113686695676520543, 2.43687262764253220589504777569, 3.45137754740827241561592182715, 4.36414772300130078373076281064, 5.38094305432879129202478260369, 5.99066425361125673322565969009, 7.16288905228318448190789636938, 7.61390269976383885758839905506, 8.545843733685224632714726384741
