L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s + 6·11-s + 3·13-s − 4·16-s − 4·17-s − 19-s + 2·20-s + 12·22-s + 4·23-s + 25-s + 6·26-s + 8·29-s − 31-s − 8·32-s − 8·34-s + 7·37-s − 2·38-s − 6·41-s + 43-s + 12·44-s + 8·46-s + 2·47-s + 2·50-s + 6·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 1.80·11-s + 0.832·13-s − 16-s − 0.970·17-s − 0.229·19-s + 0.447·20-s + 2.55·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s + 1.48·29-s − 0.179·31-s − 1.41·32-s − 1.37·34-s + 1.15·37-s − 0.324·38-s − 0.937·41-s + 0.152·43-s + 1.80·44-s + 1.17·46-s + 0.291·47-s + 0.282·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.497957345\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.497957345\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−8.991030957109522459325859295332, −8.474752266922082470750006713324, −6.95391958280063376233690841407, −6.48904292598184087272451079762, −5.95625869369177509868841929024, −4.89113030225612730960424377859, −4.22916994325197318290776181669, −3.50292231247963867488582193606, −2.49266372765463851121207350921, −1.26304483824458939740098469802,
1.26304483824458939740098469802, 2.49266372765463851121207350921, 3.50292231247963867488582193606, 4.22916994325197318290776181669, 4.89113030225612730960424377859, 5.95625869369177509868841929024, 6.48904292598184087272451079762, 6.95391958280063376233690841407, 8.474752266922082470750006713324, 8.991030957109522459325859295332