| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 6·13-s − 6·17-s − 8·19-s + 4·21-s − 27-s − 6·29-s + 33-s − 6·37-s + 6·39-s − 10·41-s + 8·43-s + 9·49-s + 6·51-s − 6·53-s + 8·57-s + 4·59-s − 2·61-s − 4·63-s + 12·67-s − 8·71-s − 2·73-s + 4·77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 1.45·17-s − 1.83·19-s + 0.872·21-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.986·37-s + 0.960·39-s − 1.56·41-s + 1.21·43-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.256·61-s − 0.503·63-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 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
−6.91243348572659873459197337224, −6.77114453439511230036963943917, −5.98633093606997687788710606516, −5.15844403752174915099379488888, −4.43574802067698234038978674358, −3.68567392097867542340724775466, −2.62166115943425452399518566435, −2.01172862456584052085901183106, 0, 0,
2.01172862456584052085901183106, 2.62166115943425452399518566435, 3.68567392097867542340724775466, 4.43574802067698234038978674358, 5.15844403752174915099379488888, 5.98633093606997687788710606516, 6.77114453439511230036963943917, 6.91243348572659873459197337224
