| L(s) = 1 | + 6·2-s + 4·4-s + 6·5-s − 168·8-s + 36·10-s + 564·11-s − 638·13-s − 1.13e3·16-s + 882·17-s + 556·19-s + 24·20-s + 3.38e3·22-s + 840·23-s − 3.08e3·25-s − 3.82e3·26-s − 4.63e3·29-s − 4.40e3·31-s − 1.44e3·32-s + 5.29e3·34-s − 2.41e3·37-s + 3.33e3·38-s − 1.00e3·40-s − 6.87e3·41-s + 9.64e3·43-s + 2.25e3·44-s + 5.04e3·46-s − 1.86e4·47-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 0.107·5-s − 0.928·8-s + 0.113·10-s + 1.40·11-s − 1.04·13-s − 1.10·16-s + 0.740·17-s + 0.353·19-s + 0.0134·20-s + 1.49·22-s + 0.331·23-s − 0.988·25-s − 1.11·26-s − 1.02·29-s − 0.822·31-s − 0.248·32-s + 0.785·34-s − 0.289·37-s + 0.374·38-s − 0.0996·40-s − 0.638·41-s + 0.795·43-s + 0.175·44-s + 0.351·46-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 11 | \( 1 - 564 T + p^{5} T^{2} \) |
| 13 | \( 1 + 638 T + p^{5} T^{2} \) |
| 17 | \( 1 - 882 T + p^{5} T^{2} \) |
| 19 | \( 1 - 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 840 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4638 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4400 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2410 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6870 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9644 T + p^{5} T^{2} \) |
| 47 | \( 1 + 18672 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33750 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18084 T + p^{5} T^{2} \) |
| 61 | \( 1 + 39758 T + p^{5} T^{2} \) |
| 67 | \( 1 + 23068 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 41110 T + p^{5} T^{2} \) |
| 79 | \( 1 - 21920 T + p^{5} T^{2} \) |
| 83 | \( 1 - 82452 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94086 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49442 T + p^{5} T^{2} \) |
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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
−9.558297844879735044680238720556, −9.313085093212430613792908526044, −7.88854648130614573677907974130, −6.82162029569779115809225730085, −5.86190432912832278467750214709, −4.99686528036810143531970372324, −3.98514492916380631155184834014, −3.13615961573469705053921326562, −1.65880678231076976581179004604, 0,
1.65880678231076976581179004604, 3.13615961573469705053921326562, 3.98514492916380631155184834014, 4.99686528036810143531970372324, 5.86190432912832278467750214709, 6.82162029569779115809225730085, 7.88854648130614573677907974130, 9.313085093212430613792908526044, 9.558297844879735044680238720556
