| L(s) = 1 | + 5-s − 2·7-s + 11-s − 4·13-s + 2·17-s + 23-s − 4·25-s + 7·31-s − 2·35-s − 3·37-s + 8·41-s + 6·43-s − 8·47-s − 3·49-s − 6·53-s + 55-s + 5·59-s − 12·61-s − 4·65-s + 7·67-s + 3·71-s + 4·73-s − 2·77-s − 10·79-s − 6·83-s + 2·85-s − 15·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s − 1.10·13-s + 0.485·17-s + 0.208·23-s − 4/5·25-s + 1.25·31-s − 0.338·35-s − 0.493·37-s + 1.24·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.134·55-s + 0.650·59-s − 1.53·61-s − 0.496·65-s + 0.855·67-s + 0.356·71-s + 0.468·73-s − 0.227·77-s − 1.12·79-s − 0.658·83-s + 0.216·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.66052303849376615023391483912, −6.91457629612246442653144098979, −6.28005608278505881970642261392, −5.63121487762866451184691812032, −4.82131397659893910760656805810, −4.03375715080835343050434711466, −3.07328878728004487658117213803, −2.43077728831070978902120289570, −1.31939690878717253102283865453, 0,
1.31939690878717253102283865453, 2.43077728831070978902120289570, 3.07328878728004487658117213803, 4.03375715080835343050434711466, 4.82131397659893910760656805810, 5.63121487762866451184691812032, 6.28005608278505881970642261392, 6.91457629612246442653144098979, 7.66052303849376615023391483912
