| L(s) = 1 | − 3-s − 2·4-s − 2·7-s + 9-s − 6·11-s + 2·12-s − 5·13-s + 4·16-s − 6·17-s + 2·19-s + 2·21-s − 9·23-s − 27-s + 4·28-s − 6·29-s + 2·31-s + 6·33-s − 2·36-s + 4·37-s + 5·39-s − 11·43-s + 12·44-s + 47-s − 4·48-s − 3·49-s + 6·51-s + 10·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 1.38·13-s + 16-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 1.87·23-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.359·31-s + 1.04·33-s − 1/3·36-s + 0.657·37-s + 0.800·39-s − 1.67·43-s + 1.80·44-s + 0.145·47-s − 0.577·48-s − 3/7·49-s + 0.840·51-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.84838565860687506954358366495, −7.16944675370096633921305160030, −6.18103019740088526076414716401, −5.44293801278956199820175588686, −4.83479567799717096115611901026, −4.13735692007891235500711481953, −3.01969767268857280935746186808, −2.07663717035188596137528899900, 0, 0,
2.07663717035188596137528899900, 3.01969767268857280935746186808, 4.13735692007891235500711481953, 4.83479567799717096115611901026, 5.44293801278956199820175588686, 6.18103019740088526076414716401, 7.16944675370096633921305160030, 7.84838565860687506954358366495
