L(s) = 1 | + 2·3-s + 5-s + 9-s − 2·11-s + 2·15-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 4·27-s − 4·29-s − 31-s − 4·33-s − 8·37-s − 6·41-s − 2·43-s + 45-s − 4·51-s + 8·53-s − 2·55-s − 8·57-s + 8·59-s − 4·67-s + 8·69-s − 6·73-s + 2·75-s + 4·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 0.179·31-s − 0.696·33-s − 1.31·37-s − 0.937·41-s − 0.304·43-s + 0.149·45-s − 0.560·51-s + 1.09·53-s − 0.269·55-s − 1.05·57-s + 1.04·59-s − 0.488·67-s + 0.963·69-s − 0.702·73-s + 0.230·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.526382683\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.526382683\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.47034525067121, −13.31146449338066, −12.77256454229126, −12.24032590808159, −11.56567060309329, −11.10129807439571, −10.47400487750569, −10.19601703762750, −9.558883525027210, −8.947690080643831, −8.746947401971207, −8.300426303647206, −7.668612646151322, −7.155791313971879, −6.700622910779225, −6.042743051410995, −5.395669690874133, −5.001043440518800, −4.263185523516663, −3.626900042440264, −3.190898901929341, −2.482855679021779, −2.096485868278731, −1.522705863720687, −0.4142003543184528,
0.4142003543184528, 1.522705863720687, 2.096485868278731, 2.482855679021779, 3.190898901929341, 3.626900042440264, 4.263185523516663, 5.001043440518800, 5.395669690874133, 6.042743051410995, 6.700622910779225, 7.155791313971879, 7.668612646151322, 8.300426303647206, 8.746947401971207, 8.947690080643831, 9.558883525027210, 10.19601703762750, 10.47400487750569, 11.10129807439571, 11.56567060309329, 12.24032590808159, 12.77256454229126, 13.31146449338066, 13.47034525067121