L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 6·11-s + 4·13-s − 15-s + 6·17-s + 6·19-s − 21-s − 4·23-s + 25-s − 27-s − 2·29-s − 2·31-s − 6·33-s + 35-s + 6·37-s − 4·39-s + 6·41-s − 12·43-s + 45-s + 49-s − 6·51-s − 4·53-s + 6·55-s − 6·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.37·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 1.04·33-s + 0.169·35-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 1.82·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.354043070\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.354043070\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.652242570532032956808522273624, −7.82515110791335845709571561054, −7.07298960460316702797495533470, −6.09714302051276605334008138850, −5.86423010733076032993144731540, −4.86090875329114854986988833555, −3.90530065244054326163413930648, −3.24749989127084101356446318476, −1.60048861461951018401731468882, −1.11043913973215741596274147012,
1.11043913973215741596274147012, 1.60048861461951018401731468882, 3.24749989127084101356446318476, 3.90530065244054326163413930648, 4.86090875329114854986988833555, 5.86423010733076032993144731540, 6.09714302051276605334008138850, 7.07298960460316702797495533470, 7.82515110791335845709571561054, 8.652242570532032956808522273624