| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 18·5-s − 216·6-s − 512·8-s + 729·9-s − 144·10-s + 8.17e3·11-s + 1.72e3·12-s + 1.42e4·13-s + 486·15-s + 4.09e3·16-s + 2.14e4·17-s − 5.83e3·18-s + 5.88e3·19-s + 1.15e3·20-s − 6.53e4·22-s − 9.87e4·23-s − 1.38e4·24-s − 7.78e4·25-s − 1.13e5·26-s + 1.96e4·27-s + 1.65e5·29-s − 3.88e3·30-s + 2.41e5·31-s − 3.27e4·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.0643·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.0455·10-s + 1.85·11-s + 0.288·12-s + 1.79·13-s + 0.0371·15-s + 1/4·16-s + 1.05·17-s − 0.235·18-s + 0.196·19-s + 0.0321·20-s − 1.30·22-s − 1.69·23-s − 0.204·24-s − 0.995·25-s − 1.27·26-s + 0.192·27-s + 1.25·29-s − 0.0262·30-s + 1.45·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.720668260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720668260\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 18 T + p^{7} T^{2} \) |
| 11 | \( 1 - 8172 T + p^{7} T^{2} \) |
| 13 | \( 1 - 14242 T + p^{7} T^{2} \) |
| 17 | \( 1 - 21462 T + p^{7} T^{2} \) |
| 19 | \( 1 - 5884 T + p^{7} T^{2} \) |
| 23 | \( 1 + 98784 T + p^{7} T^{2} \) |
| 29 | \( 1 - 165174 T + p^{7} T^{2} \) |
| 31 | \( 1 - 241312 T + p^{7} T^{2} \) |
| 37 | \( 1 - 185438 T + p^{7} T^{2} \) |
| 41 | \( 1 + 59682 T + p^{7} T^{2} \) |
| 43 | \( 1 + 809308 T + p^{7} T^{2} \) |
| 47 | \( 1 + 942096 T + p^{7} T^{2} \) |
| 53 | \( 1 - 226398 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2205732 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1156690 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3740404 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2593296 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1038742 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2280032 T + p^{7} T^{2} \) |
| 83 | \( 1 - 283404 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5227230 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6168770 T + p^{7} T^{2} \) |
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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
−10.20469563771074407083015823322, −9.625662773121024660849698049800, −8.524603203471837991108608791334, −8.078800172039724744198107843986, −6.63109656882668245115689214363, −6.02924582768812313915111455846, −4.11571939084227571707929134968, −3.30751852104742682794155727624, −1.70166152361807475035783182085, −0.976390682087626548943334349174,
0.976390682087626548943334349174, 1.70166152361807475035783182085, 3.30751852104742682794155727624, 4.11571939084227571707929134968, 6.02924582768812313915111455846, 6.63109656882668245115689214363, 8.078800172039724744198107843986, 8.524603203471837991108608791334, 9.625662773121024660849698049800, 10.20469563771074407083015823322
