| L(s) = 1 | + 3-s + 9-s − 4·11-s + 6·13-s + 6·17-s + 8·19-s − 4·23-s + 27-s + 10·29-s − 4·33-s + 6·37-s + 6·39-s − 6·41-s + 43-s + 12·47-s − 7·49-s + 6·51-s + 6·53-s + 8·57-s + 4·59-s − 6·61-s + 4·67-s − 4·69-s − 8·71-s + 10·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s + 1.83·19-s − 0.834·23-s + 0.192·27-s + 1.85·29-s − 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s + 0.152·43-s + 1.75·47-s − 49-s + 0.840·51-s + 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.768·61-s + 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.256653050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.256653050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.25334165810763, −13.86053434048880, −13.71645949215626, −13.04243393650398, −12.47177228456956, −11.94469874193214, −11.48516971341421, −10.76914935398735, −10.23874915496893, −9.922688528858054, −9.334006155698718, −8.544467102473249, −8.238396519809237, −7.723388587504086, −7.292912735840633, −6.475751473359281, −5.810267679132789, −5.458194675550600, −4.741884573171524, −3.995031867895881, −3.279625946356175, −3.031561825766237, −2.206438564032547, −1.219465026012271, −0.8085781758914221,
0.8085781758914221, 1.219465026012271, 2.206438564032547, 3.031561825766237, 3.279625946356175, 3.995031867895881, 4.741884573171524, 5.458194675550600, 5.810267679132789, 6.475751473359281, 7.292912735840633, 7.723388587504086, 8.238396519809237, 8.544467102473249, 9.334006155698718, 9.922688528858054, 10.23874915496893, 10.76914935398735, 11.48516971341421, 11.94469874193214, 12.47177228456956, 13.04243393650398, 13.71645949215626, 13.86053434048880, 14.25334165810763
