L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 5·11-s + 16-s + 4·17-s + 8·19-s − 20-s + 5·22-s + 4·23-s − 4·25-s + 5·29-s + 3·31-s − 32-s − 4·34-s − 4·37-s − 8·38-s + 40-s + 2·43-s − 5·44-s − 4·46-s + 6·47-s + 4·50-s + 9·53-s + 5·55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.50·11-s + 1/4·16-s + 0.970·17-s + 1.83·19-s − 0.223·20-s + 1.06·22-s + 0.834·23-s − 4/5·25-s + 0.928·29-s + 0.538·31-s − 0.176·32-s − 0.685·34-s − 0.657·37-s − 1.29·38-s + 0.158·40-s + 0.304·43-s − 0.753·44-s − 0.589·46-s + 0.875·47-s + 0.565·50-s + 1.23·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9654242689\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9654242689\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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
−10.14278623156165007258391762078, −9.373692494308660996484764926308, −8.306152149598525312177887687606, −7.69780043471093435649583790995, −7.08055487724091789981125828654, −5.71619227739785613107965984191, −5.01953856335493271435920798157, −3.49120843004488869714011425140, −2.57919957316952716799921143645, −0.880626761328028180035180249034,
0.880626761328028180035180249034, 2.57919957316952716799921143645, 3.49120843004488869714011425140, 5.01953856335493271435920798157, 5.71619227739785613107965984191, 7.08055487724091789981125828654, 7.69780043471093435649583790995, 8.306152149598525312177887687606, 9.373692494308660996484764926308, 10.14278623156165007258391762078