| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s − 2·14-s + 16-s − 2·17-s + 6·19-s + 2·20-s + 4·23-s − 25-s − 2·28-s + 10·29-s − 10·31-s + 32-s − 2·34-s − 4·35-s + 8·37-s + 6·38-s + 2·40-s + 10·41-s − 4·43-s + 4·46-s + 12·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.834·23-s − 1/5·25-s − 0.377·28-s + 1.85·29-s − 1.79·31-s + 0.176·32-s − 0.342·34-s − 0.676·35-s + 1.31·37-s + 0.973·38-s + 0.316·40-s + 1.56·41-s − 0.609·43-s + 0.589·46-s + 1.75·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377222362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377222362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.959503311860706269607722672745, −7.73378898932464074552193129263, −7.05895265321083254426224048131, −6.27350258758939457604625436737, −5.70319207261470080617238192214, −4.95975808822596780108243049076, −3.98742330941435803020238721272, −3.02995086886666391737100473760, −2.34513957051456022215968764679, −1.05111191786649945850127259101,
1.05111191786649945850127259101, 2.34513957051456022215968764679, 3.02995086886666391737100473760, 3.98742330941435803020238721272, 4.95975808822596780108243049076, 5.70319207261470080617238192214, 6.27350258758939457604625436737, 7.05895265321083254426224048131, 7.73378898932464074552193129263, 8.959503311860706269607722672745
