| L(s) = 1 | − 11·2-s + 89·4-s − 627·8-s + 76·11-s + 4.04e3·16-s − 836·22-s + 4.95e3·23-s − 3.12e3·25-s − 7.28e3·29-s − 2.44e4·32-s − 8.88e3·37-s + 1.17e4·43-s + 6.76e3·44-s − 5.44e4·46-s + 3.43e4·50-s − 2.45e4·53-s + 8.01e4·58-s + 1.39e5·64-s + 6.93e4·67-s + 2.22e3·71-s + 9.77e4·74-s + 8.01e4·79-s − 1.29e5·86-s − 4.76e4·88-s + 4.40e5·92-s − 2.78e5·100-s + 2.70e5·106-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.78·4-s − 3.46·8-s + 0.189·11-s + 3.95·16-s − 0.368·22-s + 1.95·23-s − 25-s − 1.60·29-s − 4.22·32-s − 1.06·37-s + 0.968·43-s + 0.526·44-s − 3.79·46-s + 1.94·50-s − 1.20·53-s + 3.12·58-s + 4.26·64-s + 1.88·67-s + 0.0523·71-s + 2.07·74-s + 1.44·79-s − 1.88·86-s − 0.655·88-s + 5.42·92-s − 2.78·100-s + 2.33·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 11 T + p^{5} T^{2} \) |
| 5 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 - 76 T + p^{5} T^{2} \) |
| 13 | \( 1 + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 - 4952 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7282 T + p^{5} T^{2} \) |
| 31 | \( 1 + p^{5} T^{2} \) |
| 37 | \( 1 + 8886 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 - 11748 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + 24550 T + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + p^{5} T^{2} \) |
| 67 | \( 1 - 69364 T + p^{5} T^{2} \) |
| 71 | \( 1 - 2224 T + p^{5} T^{2} \) |
| 73 | \( 1 + p^{5} T^{2} \) |
| 79 | \( 1 - 80168 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + p^{5} T^{2} \) |
| 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
−9.550255366381160969867957058373, −9.156662431591949536526275608152, −8.152686373300926948644608764287, −7.34843659534065527948588249252, −6.58655060304073121121350194867, −5.44783047269685256139659023580, −3.46538043135167707891574985563, −2.22861911413953242114506158430, −1.16725023901586829333008032763, 0,
1.16725023901586829333008032763, 2.22861911413953242114506158430, 3.46538043135167707891574985563, 5.44783047269685256139659023580, 6.58655060304073121121350194867, 7.34843659534065527948588249252, 8.152686373300926948644608764287, 9.156662431591949536526275608152, 9.550255366381160969867957058373
