| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s + 4·13-s + 2·17-s − 4·19-s − 21-s + 7·23-s − 27-s − 9·29-s − 2·31-s − 33-s + 37-s − 4·39-s + 8·41-s − 9·43-s + 4·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s + 4·61-s + 63-s + 9·67-s − 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1.45·23-s − 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.174·33-s + 0.164·37-s − 0.640·39-s + 1.24·41-s − 1.37·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s + 0.512·61-s + 0.125·63-s + 1.09·67-s − 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.616408142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616408142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.035948737829307169097101668566, −8.418506465353995802148045474308, −7.45176657760196889743833301470, −6.73489749812111319905027858179, −5.87732388436887148920245877641, −5.23269248499580279751040136210, −4.20919167219552969919961414854, −3.44277124885466421859584388047, −2.01848888631742411204917533466, −0.902461275409340837649759250213,
0.902461275409340837649759250213, 2.01848888631742411204917533466, 3.44277124885466421859584388047, 4.20919167219552969919961414854, 5.23269248499580279751040136210, 5.87732388436887148920245877641, 6.73489749812111319905027858179, 7.45176657760196889743833301470, 8.418506465353995802148045474308, 9.035948737829307169097101668566
