L(s) = 1 | − 726·5-s − 3.05e3i·7-s − 1.32e4i·11-s + 3.90e4·13-s + 6.58e4·17-s − 1.30e5i·19-s + 5.02e5i·23-s + 1.36e5·25-s − 2.02e5·29-s − 1.19e6i·31-s + 2.21e6i·35-s − 1.87e6·37-s − 3.09e6·41-s − 2.26e6i·43-s − 6.35e6i·47-s + ⋯ |
L(s) = 1 | − 1.16·5-s − 1.27i·7-s − 0.907i·11-s + 1.36·13-s + 0.787·17-s − 0.999i·19-s + 1.79i·23-s + 0.349·25-s − 0.285·29-s − 1.29i·31-s + 1.47i·35-s − 1.00·37-s − 1.09·41-s − 0.662i·43-s − 1.30i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7175438406\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7175438406\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 726T + 3.90e5T^{2} \) |
| 7 | \( 1 + 3.05e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.32e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.90e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 6.58e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.30e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 5.02e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 2.02e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.19e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.87e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.09e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.26e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.35e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.06e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 5.76e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.71e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.74e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.98e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.32e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.82e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.78e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.66e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 7.39e7T + 7.83e15T^{2} \) |
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
−11.23588028393465378454250933998, −10.22760799966659563517862059299, −8.785302677587838440953240196590, −7.81770812513077481574615372421, −6.99894372701087408169432808351, −5.54566028193437022397373300271, −3.91848734507428364204258636077, −3.47396882675293710578096509076, −1.18703710312566985900498957512, −0.20706452938897119757122387540,
1.52895582506519606598232654234, 3.07669278746858495999326622779, 4.23190489262732324403069674012, 5.55663056819985722961458157232, 6.76856177848605040873031603738, 8.131624285403376019852162878292, 8.681876505504639995946102334657, 10.07991383198158764479110911142, 11.21474285187317208649225378273, 12.24332434150722792388088570233