| L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·13-s + 6·17-s − 4·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s − 6·29-s − 8·31-s − 10·37-s − 4·39-s − 6·41-s + 8·43-s + 6·47-s − 3·49-s − 6·51-s + 4·57-s − 8·61-s + 2·63-s + 4·67-s + 6·69-s − 6·71-s − 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.529·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.712·71-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.72818232640287096583854635682, −7.15001151228072728345472327672, −6.01435881870512402755326279246, −5.77859618909253610703452808610, −4.97589384532422545853514681972, −3.94115256297343253062160005188, −3.56342125758668518106535855673, −2.02292253942257951840802942475, −1.42670529206492309486707433636, 0,
1.42670529206492309486707433636, 2.02292253942257951840802942475, 3.56342125758668518106535855673, 3.94115256297343253062160005188, 4.97589384532422545853514681972, 5.77859618909253610703452808610, 6.01435881870512402755326279246, 7.15001151228072728345472327672, 7.72818232640287096583854635682
