| L(s) = 1 | − 2·5-s + 4·7-s + 11-s + 2·13-s + 2·17-s − 8·23-s − 25-s − 6·29-s − 8·31-s − 8·35-s − 6·37-s + 2·41-s − 8·47-s + 9·49-s + 6·53-s − 2·55-s − 4·59-s − 6·61-s − 4·65-s + 4·67-s − 14·73-s + 4·77-s − 4·79-s + 12·83-s − 4·85-s + 6·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 1.35·35-s − 0.986·37-s + 0.312·41-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s − 1.63·73-s + 0.455·77-s − 0.450·79-s + 1.31·83-s − 0.433·85-s + 0.635·89-s + 0.838·91-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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.66051509627770747286574936361, −7.33128376884448727388193843826, −6.16366375271443718987965211775, −5.51322613332559214103852304334, −4.73618217091200823321884657659, −3.93285377374688399791870451173, −3.52453918452754346704207297036, −2.06248411406244788648042527132, −1.44422961659300135266528370838, 0,
1.44422961659300135266528370838, 2.06248411406244788648042527132, 3.52453918452754346704207297036, 3.93285377374688399791870451173, 4.73618217091200823321884657659, 5.51322613332559214103852304334, 6.16366375271443718987965211775, 7.33128376884448727388193843826, 7.66051509627770747286574936361
