| L(s) = 1 | − 3·2-s − 3·3-s + 4-s − 18·5-s + 9·6-s + 7·7-s + 21·8-s + 9·9-s + 54·10-s − 36·11-s − 3·12-s − 34·13-s − 21·14-s + 54·15-s − 71·16-s + 42·17-s − 27·18-s − 124·19-s − 18·20-s − 21·21-s + 108·22-s − 63·24-s + 199·25-s + 102·26-s − 27·27-s + 7·28-s + 102·29-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s − 1.60·5-s + 0.612·6-s + 0.377·7-s + 0.928·8-s + 1/3·9-s + 1.70·10-s − 0.986·11-s − 0.0721·12-s − 0.725·13-s − 0.400·14-s + 0.929·15-s − 1.10·16-s + 0.599·17-s − 0.353·18-s − 1.49·19-s − 0.201·20-s − 0.218·21-s + 1.04·22-s − 0.535·24-s + 1.59·25-s + 0.769·26-s − 0.192·27-s + 0.0472·28-s + 0.653·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 318 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 132 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 - 92 T + p^{3} T^{2} \) |
| 71 | \( 1 + 720 T + p^{3} T^{2} \) |
| 73 | \( 1 + 502 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 + 204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 354 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−17.11626101801103868663808827808, −16.14418605050144376207476547610, −14.91290151933212586147181506316, −12.75890770904133744320234829033, −11.38988080428046421269273778849, −10.29314165331111143855582291138, −8.366976436165780451630990829562, −7.43205058596234115633185970014, −4.59505455463720251261035786388, 0,
4.59505455463720251261035786388, 7.43205058596234115633185970014, 8.366976436165780451630990829562, 10.29314165331111143855582291138, 11.38988080428046421269273778849, 12.75890770904133744320234829033, 14.91290151933212586147181506316, 16.14418605050144376207476547610, 17.11626101801103868663808827808
