L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s − 6·11-s − 4·13-s + 2·14-s + 16-s + 6·17-s − 4·19-s − 20-s − 6·22-s + 25-s − 4·26-s + 2·28-s + 6·29-s − 4·31-s + 32-s + 6·34-s − 2·35-s + 8·37-s − 4·38-s − 40-s + 8·43-s − 6·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.80·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.784·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.338·35-s + 1.31·37-s − 0.648·38-s − 0.158·40-s + 1.21·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.323698086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323698086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.26837603536780067776326433213, −12.93373489144007201814533639274, −12.20577425790790506710344163162, −11.00597902346516808155584438017, −10.07339932784564217269318227134, −8.137806933221736892131373840887, −7.42257774366004622063287945506, −5.57449936126795434832243724178, −4.56211949870166416184694202500, −2.68907161640043859094582658785,
2.68907161640043859094582658785, 4.56211949870166416184694202500, 5.57449936126795434832243724178, 7.42257774366004622063287945506, 8.137806933221736892131373840887, 10.07339932784564217269318227134, 11.00597902346516808155584438017, 12.20577425790790506710344163162, 12.93373489144007201814533639274, 14.26837603536780067776326433213