| L(s) = 1 | − 2·5-s + 17-s + 2·19-s + 4·23-s − 25-s − 6·29-s + 8·31-s + 2·37-s − 10·41-s + 4·43-s + 8·47-s − 8·53-s + 4·59-s − 6·61-s + 4·67-s + 12·71-s + 10·73-s + 4·79-s + 12·83-s − 2·85-s − 6·89-s − 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 1.09·53-s + 0.520·59-s − 0.768·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.450·79-s + 1.31·83-s − 0.216·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.79353795213903, −13.34198843518672, −12.82758338438904, −12.19516749310018, −11.94420980818498, −11.44916583650655, −10.90776567692231, −10.56615744855119, −9.847860716252344, −9.375408078032698, −8.994739553287027, −8.193857015899127, −7.901364872904389, −7.543622724235403, −6.678443499043390, −6.594258218221519, −5.641760442718006, −5.199504903870582, −4.688955887001083, −3.891406753430815, −3.690389751320643, −2.911633420093172, −2.380767327874189, −1.468405776844077, −0.8180080597540304, 0,
0.8180080597540304, 1.468405776844077, 2.380767327874189, 2.911633420093172, 3.690389751320643, 3.891406753430815, 4.688955887001083, 5.199504903870582, 5.641760442718006, 6.594258218221519, 6.678443499043390, 7.543622724235403, 7.901364872904389, 8.193857015899127, 8.994739553287027, 9.375408078032698, 9.847860716252344, 10.56615744855119, 10.90776567692231, 11.44916583650655, 11.94420980818498, 12.19516749310018, 12.82758338438904, 13.34198843518672, 13.79353795213903
