| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 3·11-s + 13-s − 5·17-s − 8·19-s + 3·21-s + 2·23-s + 9·27-s − 29-s − 2·31-s + 9·33-s + 10·37-s + 3·39-s − 6·41-s − 4·43-s + 11·47-s + 49-s − 15·51-s + 6·53-s − 24·57-s − 10·59-s + 6·63-s − 10·67-s + 6·69-s − 10·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 0.904·11-s + 0.277·13-s − 1.21·17-s − 1.83·19-s + 0.654·21-s + 0.417·23-s + 1.73·27-s − 0.185·29-s − 0.359·31-s + 1.56·33-s + 1.64·37-s + 0.480·39-s − 0.937·41-s − 0.609·43-s + 1.60·47-s + 1/7·49-s − 2.10·51-s + 0.824·53-s − 3.17·57-s − 1.30·59-s + 0.755·63-s − 1.22·67-s + 0.722·69-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.800932328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800932328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−10.33350136132695345505774184140, −9.129359814341148121145330711601, −8.878316880045985327616196183506, −8.074771231686246722602580121512, −7.11680708466198240902328128179, −6.24807203668297539781117592997, −4.48739889783168415514429250228, −3.91658437283551739137960947678, −2.64505632772490851790668490036, −1.71009994790434979397954725806,
1.71009994790434979397954725806, 2.64505632772490851790668490036, 3.91658437283551739137960947678, 4.48739889783168415514429250228, 6.24807203668297539781117592997, 7.11680708466198240902328128179, 8.074771231686246722602580121512, 8.878316880045985327616196183506, 9.129359814341148121145330711601, 10.33350136132695345505774184140
