| L(s) = 1 | − 3-s − 7-s + 9-s + 2·11-s − 13-s − 6·17-s + 8·19-s + 21-s + 4·23-s − 5·25-s − 27-s − 6·29-s − 4·31-s − 2·33-s − 2·37-s + 39-s − 4·43-s + 6·47-s + 49-s + 6·51-s + 6·53-s − 8·57-s − 10·59-s + 10·61-s − 63-s + 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 1.45·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s − 25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 1.05·57-s − 1.30·59-s + 1.28·61-s − 0.125·63-s + 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.86338375420546656978466574081, −7.07895522129620060589940906087, −6.70223035305728523216116762898, −5.66578639774051390411165534916, −5.21875715958066097976881878268, −4.17962728559360354332821031426, −3.51232446979697290516032970963, −2.39110741933362874392361674478, −1.29735345052249992741707860988, 0,
1.29735345052249992741707860988, 2.39110741933362874392361674478, 3.51232446979697290516032970963, 4.17962728559360354332821031426, 5.21875715958066097976881878268, 5.66578639774051390411165534916, 6.70223035305728523216116762898, 7.07895522129620060589940906087, 7.86338375420546656978466574081
