L(s) = 1 | − 32·4-s + 25·7-s + 775·13-s + 1.02e3·16-s − 1.71e3·19-s − 800·28-s + 2.72e3·31-s − 1.65e4·37-s − 2.24e4·43-s − 1.61e4·49-s − 2.48e4·52-s + 5.69e4·61-s − 3.27e4·64-s − 7.34e4·67-s + 1.45e3·73-s + 5.47e4·76-s − 1.00e5·79-s + 1.93e4·91-s − 1.77e5·97-s − 1.40e5·103-s + 1.33e5·109-s + 2.56e4·112-s + ⋯ |
L(s) = 1 | − 4-s + 0.192·7-s + 1.27·13-s + 16-s − 1.08·19-s − 0.192·28-s + 0.508·31-s − 1.98·37-s − 1.85·43-s − 0.962·49-s − 1.27·52-s + 1.95·61-s − 64-s − 1.99·67-s + 0.0318·73-s + 1.08·76-s − 1.81·79-s + 0.245·91-s − 1.91·97-s − 1.30·103-s + 1.07·109-s + 0.192·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p^{5} T^{2} \) |
| 7 | \( 1 - 25 T + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 - 775 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 + 1711 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 - 2723 T + p^{5} T^{2} \) |
| 37 | \( 1 + 16550 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 + 22475 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 - 56927 T + p^{5} T^{2} \) |
| 67 | \( 1 + 73475 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 - 1450 T + p^{5} T^{2} \) |
| 79 | \( 1 + 100564 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + 177725 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77995784373186041546310499609, −9.925979660163972277411813403660, −8.692624502105831220640237477844, −8.295656358778636899901034091369, −6.74304981344413844976704359070, −5.57462453937571278724762066698, −4.44723279043932538266830743420, −3.40797398115610612264803115084, −1.50201811755038771010386861744, 0,
1.50201811755038771010386861744, 3.40797398115610612264803115084, 4.44723279043932538266830743420, 5.57462453937571278724762066698, 6.74304981344413844976704359070, 8.295656358778636899901034091369, 8.692624502105831220640237477844, 9.925979660163972277411813403660, 10.77995784373186041546310499609