| L(s) = 1 | + 1.25e3·7-s + 2.00e3·13-s − 4.30e4·19-s − 7.81e4·25-s − 1.78e5·31-s + 3.35e5·37-s − 1.03e6·43-s + 7.51e5·49-s + 1.99e6·61-s − 4.44e6·67-s + 5.03e6·73-s + 4.51e6·79-s + 2.52e6·91-s − 1.75e7·97-s − 1.38e7·103-s − 1.68e7·109-s + ⋯ |
| L(s) = 1 | + 1.38·7-s + 0.253·13-s − 1.44·19-s − 25-s − 1.07·31-s + 1.08·37-s − 1.98·43-s + 0.912·49-s + 1.12·61-s − 1.80·67-s + 1.51·73-s + 1.03·79-s + 0.350·91-s − 1.94·97-s − 1.25·103-s − 1.24·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{7} T^{2} \) |
| 7 | \( 1 - 1255 T + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 - 2009 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 + 43091 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 + 178916 T + p^{7} T^{2} \) |
| 37 | \( 1 - 335663 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 + 1035224 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 - 1998347 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4443527 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 - 5038001 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4517617 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 + 17521555 T + p^{7} T^{2} \) |
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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
−9.502627055334944350784045419311, −8.428280797017265751936247294506, −7.916815332055127740899536611003, −6.75742616438303758030786838093, −5.67123539067613384059564221637, −4.69917422912118166777161973548, −3.80160520539788019924963058764, −2.25979129300032754394409917584, −1.43384898757086863472407572436, 0,
1.43384898757086863472407572436, 2.25979129300032754394409917584, 3.80160520539788019924963058764, 4.69917422912118166777161973548, 5.67123539067613384059564221637, 6.75742616438303758030786838093, 7.916815332055127740899536611003, 8.428280797017265751936247294506, 9.502627055334944350784045419311
