| L(s) = 1 | − 3·3-s + 2·5-s + 4·7-s + 6·9-s + 2·11-s + 5·13-s − 6·15-s + 4·17-s − 2·19-s − 12·21-s − 23-s − 25-s − 9·27-s + 7·29-s + 3·31-s − 6·33-s + 8·35-s − 2·37-s − 15·39-s − 9·41-s − 8·43-s + 12·45-s − 9·47-s + 9·49-s − 12·51-s − 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 1.51·7-s + 2·9-s + 0.603·11-s + 1.38·13-s − 1.54·15-s + 0.970·17-s − 0.458·19-s − 2.61·21-s − 0.208·23-s − 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.538·31-s − 1.04·33-s + 1.35·35-s − 0.328·37-s − 2.40·39-s − 1.40·41-s − 1.21·43-s + 1.78·45-s − 1.31·47-s + 9/7·49-s − 1.68·51-s − 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553218770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553218770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.866992454756712207798623866216, −8.608238956261300097073169258146, −7.940995723732562677345333914110, −6.56889089382194248049775428165, −6.29658206116634366505284597932, −5.28615499787164158880149394506, −4.88899916461674802073833414510, −3.75737483622489943039302637500, −1.75965596937335757332008532480, −1.11834268237837652771743629830,
1.11834268237837652771743629830, 1.75965596937335757332008532480, 3.75737483622489943039302637500, 4.88899916461674802073833414510, 5.28615499787164158880149394506, 6.29658206116634366505284597932, 6.56889089382194248049775428165, 7.940995723732562677345333914110, 8.608238956261300097073169258146, 9.866992454756712207798623866216
