| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 6·13-s − 6·17-s − 4·19-s − 21-s + 4·23-s + 27-s − 2·29-s + 8·31-s − 4·33-s − 6·37-s + 6·39-s + 6·41-s − 8·43-s + 49-s − 6·51-s − 6·53-s − 4·57-s − 4·59-s + 10·61-s − 63-s − 8·67-s + 4·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.960·39-s + 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.125·63-s − 0.977·67-s + 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112139645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112139645\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.64919869180238, −15.56729685031153, −14.94114938853711, −14.14651653484397, −13.45736139130297, −13.27251678074427, −12.85511338006070, −12.09824436051169, −11.16993507564626, −10.87902369177252, −10.37223077596368, −9.611579559731254, −8.961535145261365, −8.421531434320666, −8.143522009238804, −7.221305776456903, −6.532263421785268, −6.193437833963176, −5.218738937487244, −4.584469719857668, −3.849458637213848, −3.175005796702375, −2.471733187469402, −1.743744230607156, −0.5866544347963541,
0.5866544347963541, 1.743744230607156, 2.471733187469402, 3.175005796702375, 3.849458637213848, 4.584469719857668, 5.218738937487244, 6.193437833963176, 6.532263421785268, 7.221305776456903, 8.143522009238804, 8.421531434320666, 8.961535145261365, 9.611579559731254, 10.37223077596368, 10.87902369177252, 11.16993507564626, 12.09824436051169, 12.85511338006070, 13.27251678074427, 13.45736139130297, 14.14651653484397, 14.94114938853711, 15.56729685031153, 15.64919869180238
