L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s − 2·9-s + 12-s + 2·13-s − 16-s + 2·17-s − 2·18-s + 6·19-s − 3·23-s + 3·24-s + 2·26-s + 5·27-s + 7·29-s + 2·31-s + 5·32-s + 2·34-s + 2·36-s − 8·37-s + 6·38-s − 2·39-s + 5·41-s + 7·43-s − 3·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s − 2/3·9-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.37·19-s − 0.625·23-s + 0.612·24-s + 0.392·26-s + 0.962·27-s + 1.29·29-s + 0.359·31-s + 0.883·32-s + 0.342·34-s + 1/3·36-s − 1.31·37-s + 0.973·38-s − 0.320·39-s + 0.780·41-s + 1.06·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.429907065\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429907065\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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
−9.794601867468943853780558791580, −8.840417764107119650391557478811, −8.229420821213096566462125694649, −7.06349890679776787399860637895, −5.97597953257952605064373825469, −5.55596599944499063020828796378, −4.68964805412513772279409275332, −3.67058035150319672253364857138, −2.78773100105470522979565222972, −0.835384392637813802495063850636,
0.835384392637813802495063850636, 2.78773100105470522979565222972, 3.67058035150319672253364857138, 4.68964805412513772279409275332, 5.55596599944499063020828796378, 5.97597953257952605064373825469, 7.06349890679776787399860637895, 8.229420821213096566462125694649, 8.840417764107119650391557478811, 9.794601867468943853780558791580