| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 5·11-s − 6·13-s − 3·17-s − 19-s + 2·21-s + 4·23-s + 5·27-s − 6·29-s − 8·31-s + 5·33-s − 2·37-s + 6·39-s − 7·41-s − 4·43-s + 2·47-s − 3·49-s + 3·51-s − 4·53-s + 57-s − 4·59-s − 10·61-s + 4·63-s − 3·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 1.50·11-s − 1.66·13-s − 0.727·17-s − 0.229·19-s + 0.436·21-s + 0.834·23-s + 0.962·27-s − 1.11·29-s − 1.43·31-s + 0.870·33-s − 0.328·37-s + 0.960·39-s − 1.09·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s + 0.420·51-s − 0.549·53-s + 0.132·57-s − 0.520·59-s − 1.28·61-s + 0.503·63-s − 0.366·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−7.28780604728765338110068753725, −6.63866807642751258008474997429, −5.80765212604268561972240873366, −5.13020086764721657678442496191, −4.76742150038814650314621454251, −3.43468471669899402499419844296, −2.77354044748557324889948572742, −1.99286087301063042588139388962, 0, 0,
1.99286087301063042588139388962, 2.77354044748557324889948572742, 3.43468471669899402499419844296, 4.76742150038814650314621454251, 5.13020086764721657678442496191, 5.80765212604268561972240873366, 6.63866807642751258008474997429, 7.28780604728765338110068753725
