| L(s) = 1 | + 3-s − 2·5-s + 7-s − 2·9-s + 4·11-s − 13-s − 2·15-s − 5·17-s + 19-s + 21-s + 9·23-s − 25-s − 5·27-s − 5·29-s − 2·31-s + 4·33-s − 2·35-s − 10·37-s − 39-s + 12·41-s + 10·43-s + 4·45-s − 12·47-s − 6·49-s − 5·51-s − 9·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s − 1.21·17-s + 0.229·19-s + 0.218·21-s + 1.87·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s − 0.359·31-s + 0.696·33-s − 0.338·35-s − 1.64·37-s − 0.160·39-s + 1.87·41-s + 1.52·43-s + 0.596·45-s − 1.75·47-s − 6/7·49-s − 0.700·51-s − 1.23·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.925105762829320724222848680947, −7.27016102593669119698614109672, −6.67227261307543485943205898304, −5.69049761513022482382435007500, −4.81016416757398527783630820776, −4.03773310112722838344201474726, −3.39592564013928934752432431434, −2.50538477868688370744421399200, −1.42755397424825038577276994020, 0,
1.42755397424825038577276994020, 2.50538477868688370744421399200, 3.39592564013928934752432431434, 4.03773310112722838344201474726, 4.81016416757398527783630820776, 5.69049761513022482382435007500, 6.67227261307543485943205898304, 7.27016102593669119698614109672, 7.925105762829320724222848680947
