L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s + 2·13-s − 14-s + 16-s − 6·17-s + 2·19-s − 20-s − 22-s + 6·23-s + 25-s − 2·26-s + 28-s + 8·31-s − 32-s + 6·34-s − 35-s + 8·37-s − 2·38-s + 40-s − 4·43-s + 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s − 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.31·37-s − 0.324·38-s + 0.158·40-s − 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.364041629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.364041629\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.961778104760769718731558399939, −7.45309184490894129165655349329, −6.49440962604456990636621030242, −6.25367743349795758455323227377, −4.94357038776222701645227283953, −4.48854957921016093577088018823, −3.43207112803786454487926086359, −2.66952622753625006218141877321, −1.61465266032745637085546260184, −0.69023203131919387620237309728,
0.69023203131919387620237309728, 1.61465266032745637085546260184, 2.66952622753625006218141877321, 3.43207112803786454487926086359, 4.48854957921016093577088018823, 4.94357038776222701645227283953, 6.25367743349795758455323227377, 6.49440962604456990636621030242, 7.45309184490894129165655349329, 7.961778104760769718731558399939