| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 125·5-s + 216·6-s − 1.22e3·7-s + 512·8-s + 729·9-s + 1.00e3·10-s − 4.66e3·11-s + 1.72e3·12-s + 9.14e3·13-s − 9.80e3·14-s + 3.37e3·15-s + 4.09e3·16-s − 5.65e3·17-s + 5.83e3·18-s − 6.85e3·19-s + 8.00e3·20-s − 3.31e4·21-s − 3.73e4·22-s + 2.85e4·23-s + 1.38e4·24-s + 1.56e4·25-s + 7.31e4·26-s + 1.96e4·27-s − 7.84e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.35·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.05·11-s + 0.288·12-s + 1.15·13-s − 0.955·14-s + 0.258·15-s + 0.250·16-s − 0.279·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.779·21-s − 0.747·22-s + 0.489·23-s + 0.204·24-s + 0.199·25-s + 0.816·26-s + 0.192·27-s − 0.675·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 5 | \( 1 - 125T \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 7 | \( 1 + 1.22e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.14e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.65e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 2.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.14e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.57e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.66e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.73e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.27e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.01e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.85e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.56e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.47e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.13e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.73e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.37e6T + 8.07e13T^{2} \) |
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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
−9.164073978183721224904676647479, −8.341191475945057821662707134115, −7.17434124696718040243101650654, −6.36422169591260170142825422954, −5.59607989672739009182414260566, −4.39055568691307006003267090815, −3.27730108804324398844902894014, −2.74262283894865553712693231409, −1.49482749451068041985197332545, 0,
1.49482749451068041985197332545, 2.74262283894865553712693231409, 3.27730108804324398844902894014, 4.39055568691307006003267090815, 5.59607989672739009182414260566, 6.36422169591260170142825422954, 7.17434124696718040243101650654, 8.341191475945057821662707134115, 9.164073978183721224904676647479
