| L(s) = 1 | − 3-s − 2·9-s + 5·11-s − 4·13-s − 3·17-s + 19-s + 2·23-s + 5·27-s + 29-s + 2·31-s − 5·33-s + 2·37-s + 4·39-s − 41-s − 8·43-s + 2·47-s − 7·49-s + 3·51-s − 8·53-s − 57-s − 4·59-s + 14·61-s − 5·67-s − 2·69-s + 2·71-s + 15·73-s − 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1.50·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s + 0.417·23-s + 0.962·27-s + 0.185·29-s + 0.359·31-s − 0.870·33-s + 0.328·37-s + 0.640·39-s − 0.156·41-s − 1.21·43-s + 0.291·47-s − 49-s + 0.420·51-s − 1.09·53-s − 0.132·57-s − 0.520·59-s + 1.79·61-s − 0.610·67-s − 0.240·69-s + 0.237·71-s + 1.75·73-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261044061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261044061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 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
−8.187682859146392331795424570676, −7.21917067598856192748507297971, −6.58324909463027032895823559885, −6.13860995388569913742023437137, −5.10928259306151332705079550400, −4.67115781491352787850356986377, −3.67923579534677600307740346124, −2.81673271991933169865882668989, −1.79669616154167531877342025233, −0.60912750332285794345526906870,
0.60912750332285794345526906870, 1.79669616154167531877342025233, 2.81673271991933169865882668989, 3.67923579534677600307740346124, 4.67115781491352787850356986377, 5.10928259306151332705079550400, 6.13860995388569913742023437137, 6.58324909463027032895823559885, 7.21917067598856192748507297971, 8.187682859146392331795424570676
