| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 4·13-s + 14-s + 16-s − 2·19-s + 3·20-s − 6·23-s + 4·25-s + 4·26-s + 28-s − 6·29-s + 5·31-s + 32-s + 3·35-s + 2·37-s − 2·38-s + 3·40-s + 6·41-s + 10·43-s − 6·46-s + 6·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s − 1.11·29-s + 0.898·31-s + 0.176·32-s + 0.507·35-s + 0.328·37-s − 0.324·38-s + 0.474·40-s + 0.937·41-s + 1.52·43-s − 0.884·46-s + 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.843552356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.843552356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.015255053006877536003075086198, −7.12468139474762615848955209063, −6.29574961437633563574123917046, −5.84315760589664083898483470884, −5.39074241880952165115421872247, −4.30456169908575776721292993281, −3.80538107085536458846182311961, −2.58138575780524151090389938489, −2.03246862806699813641002498927, −1.09285120951784850918046733268,
1.09285120951784850918046733268, 2.03246862806699813641002498927, 2.58138575780524151090389938489, 3.80538107085536458846182311961, 4.30456169908575776721292993281, 5.39074241880952165115421872247, 5.84315760589664083898483470884, 6.29574961437633563574123917046, 7.12468139474762615848955209063, 8.015255053006877536003075086198
