| L(s) = 1 | − 110·5-s + 504·7-s − 3.81e3·11-s + 9.57e3·13-s − 2.60e4·17-s − 3.83e4·19-s + 7.11e4·23-s − 6.60e4·25-s − 7.42e4·29-s − 2.75e5·31-s − 5.54e4·35-s − 2.66e5·37-s − 6.84e5·41-s + 2.45e5·43-s − 4.78e5·47-s − 5.69e5·49-s + 5.69e5·53-s + 4.19e5·55-s + 1.52e6·59-s − 2.64e6·61-s − 1.05e6·65-s + 1.41e6·67-s + 3.51e6·71-s + 4.73e6·73-s − 1.92e6·77-s + 4.66e6·79-s + 5.72e6·83-s + ⋯ |
| L(s) = 1 | − 0.393·5-s + 0.555·7-s − 0.863·11-s + 1.20·13-s − 1.28·17-s − 1.28·19-s + 1.21·23-s − 0.845·25-s − 0.565·29-s − 1.66·31-s − 0.218·35-s − 0.865·37-s − 1.55·41-s + 0.471·43-s − 0.672·47-s − 0.691·49-s + 0.525·53-s + 0.339·55-s + 0.966·59-s − 1.48·61-s − 0.475·65-s + 0.575·67-s + 1.16·71-s + 1.42·73-s − 0.479·77-s + 1.06·79-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 22 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 72 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 3812 T + p^{7} T^{2} \) |
| 13 | \( 1 - 9574 T + p^{7} T^{2} \) |
| 17 | \( 1 + 26098 T + p^{7} T^{2} \) |
| 19 | \( 1 + 38308 T + p^{7} T^{2} \) |
| 23 | \( 1 - 71128 T + p^{7} T^{2} \) |
| 29 | \( 1 + 74262 T + p^{7} T^{2} \) |
| 31 | \( 1 + 275680 T + p^{7} T^{2} \) |
| 37 | \( 1 + 266610 T + p^{7} T^{2} \) |
| 41 | \( 1 + 684762 T + p^{7} T^{2} \) |
| 43 | \( 1 - 245956 T + p^{7} T^{2} \) |
| 47 | \( 1 + 478800 T + p^{7} T^{2} \) |
| 53 | \( 1 - 569410 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1525324 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2640458 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1416236 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3511304 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4738618 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4661488 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5729252 T + p^{7} T^{2} \) |
| 89 | \( 1 + 11993514 T + p^{7} T^{2} \) |
| 97 | \( 1 - 7150754 T + p^{7} T^{2} \) |
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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
−12.79637177853653209299608463116, −11.25620642643203795364777386014, −10.73516447279560588597168261796, −8.976829294259475388698540698985, −8.067197902848580019607992799315, −6.68663179537156498874009517500, −5.15980414580377369855551591751, −3.76029908188727902540803641744, −1.93289758286239129558867473244, 0,
1.93289758286239129558867473244, 3.76029908188727902540803641744, 5.15980414580377369855551591751, 6.68663179537156498874009517500, 8.067197902848580019607992799315, 8.976829294259475388698540698985, 10.73516447279560588597168261796, 11.25620642643203795364777386014, 12.79637177853653209299608463116
