| L(s) = 1 | + 5-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 25-s + 10·29-s − 6·37-s − 10·41-s + 8·43-s − 4·47-s + 6·53-s + 4·55-s + 10·61-s − 2·65-s + 8·67-s + 8·71-s + 2·73-s + 4·79-s + 12·83-s − 2·85-s + 6·89-s − 4·95-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.85·29-s − 0.986·37-s − 1.56·41-s + 1.21·43-s − 0.583·47-s + 0.824·53-s + 0.539·55-s + 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.949·71-s + 0.234·73-s + 0.450·79-s + 1.31·83-s − 0.216·85-s + 0.635·89-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988140880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988140880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 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 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.55710406082953, −12.84084279794277, −12.41141490046602, −12.03037883823229, −11.56928135359093, −10.94811245180996, −10.53676286825949, −9.878726491922264, −9.735657101980314, −8.890729103883036, −8.650757064298373, −8.229350063222163, −7.408241400145190, −6.830401650363839, −6.548610376030790, −6.129961060575561, −5.347230267646434, −4.837546403640600, −4.397839782342330, −3.697725669168360, −3.248915209449942, −2.223815952393123, −2.148375756233110, −1.175895526383572, −0.5477556543709899,
0.5477556543709899, 1.175895526383572, 2.148375756233110, 2.223815952393123, 3.248915209449942, 3.697725669168360, 4.397839782342330, 4.837546403640600, 5.347230267646434, 6.129961060575561, 6.548610376030790, 6.830401650363839, 7.408241400145190, 8.229350063222163, 8.650757064298373, 8.890729103883036, 9.735657101980314, 9.878726491922264, 10.53676286825949, 10.94811245180996, 11.56928135359093, 12.03037883823229, 12.41141490046602, 12.84084279794277, 13.55710406082953
