| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 2·13-s − 15-s − 17-s − 5·19-s − 21-s − 3·23-s + 25-s − 27-s + 29-s + 4·31-s − 33-s + 35-s + 2·37-s + 2·39-s + 5·43-s + 45-s + 12·47-s + 49-s + 51-s − 11·53-s + 55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s − 0.242·17-s − 1.14·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.718·31-s − 0.174·33-s + 0.169·35-s + 0.328·37-s + 0.320·39-s + 0.762·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s − 1.51·53-s + 0.134·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.34457782939181709424084720761, −6.53583356773937172103406228461, −6.06110703871559797280887916640, −5.38993169741513287606410606229, −4.47573155439898455173737348904, −4.19309833479626174201077204070, −2.89399925945670519622419056448, −2.12262845192012560117696198341, −1.24237479064549932086685680338, 0,
1.24237479064549932086685680338, 2.12262845192012560117696198341, 2.89399925945670519622419056448, 4.19309833479626174201077204070, 4.47573155439898455173737348904, 5.38993169741513287606410606229, 6.06110703871559797280887916640, 6.53583356773937172103406228461, 7.34457782939181709424084720761
