| L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·11-s − 6·13-s + 2·17-s + 2·21-s + 4·23-s − 27-s + 8·31-s − 2·33-s − 2·37-s + 6·39-s + 2·41-s − 4·43-s + 8·47-s − 3·49-s − 2·51-s − 6·53-s + 10·59-s − 2·61-s − 2·63-s − 8·67-s − 4·69-s − 12·71-s − 4·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.436·21-s + 0.834·23-s − 0.192·27-s + 1.43·31-s − 0.348·33-s − 0.328·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s + 1.30·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s − 1.42·71-s − 0.468·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.72199070301809904349647913508, −7.10425854258391636570493599153, −6.52561295189031732145830408982, −5.76734586986262743027442362824, −4.95322240062946507868987311462, −4.32938729115317135446477209662, −3.26039161766832676110489432185, −2.50624727693911496278243716052, −1.20179606604581819027427628835, 0,
1.20179606604581819027427628835, 2.50624727693911496278243716052, 3.26039161766832676110489432185, 4.32938729115317135446477209662, 4.95322240062946507868987311462, 5.76734586986262743027442362824, 6.52561295189031732145830408982, 7.10425854258391636570493599153, 7.72199070301809904349647913508
