| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 8·23-s − 27-s + 2·29-s + 8·31-s + 4·33-s + 37-s + 2·39-s + 10·41-s + 12·43-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s + 10·61-s − 4·67-s − 8·69-s + 8·71-s + 6·73-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.164·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.488·67-s − 0.963·69-s + 0.949·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.059564219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.059564219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.94950046930758, −12.75820802610981, −12.42161010222796, −11.66322375858158, −11.11917532715277, −11.00595548721235, −10.27929856967145, −10.06889677378739, −9.420835055622108, −8.785348590522096, −8.513930125312928, −7.662612768901667, −7.491299566220211, −6.868917354888949, −6.258869844650330, −5.941258159686705, −5.215189513326125, −4.742107010860647, −4.498188551829398, −3.734730236026623, −2.869191802531595, −2.547384666642071, −1.972617280575583, −0.8623301690928723, −0.5596687341055813,
0.5596687341055813, 0.8623301690928723, 1.972617280575583, 2.547384666642071, 2.869191802531595, 3.734730236026623, 4.498188551829398, 4.742107010860647, 5.215189513326125, 5.941258159686705, 6.258869844650330, 6.868917354888949, 7.491299566220211, 7.662612768901667, 8.513930125312928, 8.785348590522096, 9.420835055622108, 10.06889677378739, 10.27929856967145, 11.00595548721235, 11.11917532715277, 11.66322375858158, 12.42161010222796, 12.75820802610981, 12.94950046930758
