L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s − 2·9-s − 10-s − 12-s + 14-s − 15-s − 16-s − 2·17-s − 2·18-s − 4·19-s + 20-s + 21-s + 8·23-s − 3·24-s + 25-s − 5·27-s − 28-s + 3·29-s − 30-s + 10·31-s + 5·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 0.962·27-s − 0.188·28-s + 0.557·29-s − 0.182·30-s + 1.79·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.159657996\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159657996\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−14.30109269778309, −13.61556099046888, −13.36773360724576, −12.90868744861101, −12.24743466565233, −11.80981409790777, −11.31664449705615, −10.81306983611795, −10.17033831758257, −9.460793214590584, −9.021642626325112, −8.458738741071171, −8.266717318750527, −7.635691207027407, −6.713617224028092, −6.442365095831044, −5.645694574887135, −5.104247263619718, −4.430327032510017, −4.272856906500729, −3.325254080719714, −2.899951214095906, −2.433410608856972, −1.328073502189051, −0.4408675677091872,
0.4408675677091872, 1.328073502189051, 2.433410608856972, 2.899951214095906, 3.325254080719714, 4.272856906500729, 4.430327032510017, 5.104247263619718, 5.645694574887135, 6.442365095831044, 6.713617224028092, 7.635691207027407, 8.266717318750527, 8.458738741071171, 9.021642626325112, 9.460793214590584, 10.17033831758257, 10.81306983611795, 11.31664449705615, 11.80981409790777, 12.24743466565233, 12.90868744861101, 13.36773360724576, 13.61556099046888, 14.30109269778309