| L(s) = 1 | + 36·3-s + 125·5-s − 776·7-s − 891·9-s + 124·11-s − 1.30e4·13-s + 4.50e3·15-s − 1.59e4·17-s + 2.05e4·19-s − 2.79e4·21-s + 2.92e4·23-s + 1.56e4·25-s − 1.10e5·27-s − 2.21e5·29-s + 1.09e5·31-s + 4.46e3·33-s − 9.70e4·35-s + 7.34e4·37-s − 4.70e5·39-s + 1.27e4·41-s − 2.90e5·43-s − 1.11e5·45-s − 1.26e6·47-s − 2.21e5·49-s − 5.74e5·51-s − 3.95e5·53-s + 1.55e4·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.447·5-s − 0.855·7-s − 0.407·9-s + 0.0280·11-s − 1.65·13-s + 0.344·15-s − 0.787·17-s + 0.686·19-s − 0.658·21-s + 0.500·23-s + 1/5·25-s − 1.08·27-s − 1.68·29-s + 0.661·31-s + 0.0216·33-s − 0.382·35-s + 0.238·37-s − 1.27·39-s + 0.0289·41-s − 0.557·43-s − 0.182·45-s − 1.78·47-s − 0.268·49-s − 0.606·51-s − 0.365·53-s + 0.0125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 - 4 p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 + 776 T + p^{7} T^{2} \) |
| 11 | \( 1 - 124 T + p^{7} T^{2} \) |
| 13 | \( 1 + 13082 T + p^{7} T^{2} \) |
| 17 | \( 1 + 15950 T + p^{7} T^{2} \) |
| 19 | \( 1 - 20516 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29224 T + p^{7} T^{2} \) |
| 29 | \( 1 + 221482 T + p^{7} T^{2} \) |
| 31 | \( 1 - 109760 T + p^{7} T^{2} \) |
| 37 | \( 1 - 73422 T + p^{7} T^{2} \) |
| 41 | \( 1 - 12762 T + p^{7} T^{2} \) |
| 43 | \( 1 + 290548 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1269152 T + p^{7} T^{2} \) |
| 53 | \( 1 + 395778 T + p^{7} T^{2} \) |
| 59 | \( 1 + 421492 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2122250 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3132868 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5376552 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4985466 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3867504 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6190196 T + p^{7} T^{2} \) |
| 89 | \( 1 - 1124394 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9968098 T + p^{7} T^{2} \) |
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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
−12.64807893876384547387021669959, −11.33023705772055506535114603085, −9.774292927455621135157787189598, −9.256811001897131496001900092966, −7.82758963286784406788785511872, −6.59144689519885099851781754421, −5.10356500691533893872930739944, −3.29366578871756918340465239756, −2.21565093762334056604785333535, 0,
2.21565093762334056604785333535, 3.29366578871756918340465239756, 5.10356500691533893872930739944, 6.59144689519885099851781754421, 7.82758963286784406788785511872, 9.256811001897131496001900092966, 9.774292927455621135157787189598, 11.33023705772055506535114603085, 12.64807893876384547387021669959
