| L(s) = 1 | + 2.32·3-s + 1.72·5-s + 45.7·7-s − 237.·9-s − 419.·11-s + 626.·13-s + 4.01·15-s − 1.14e3·17-s + 985.·19-s + 106.·21-s − 2.93e3·23-s − 3.12e3·25-s − 1.11e3·27-s − 4.57e3·29-s + 961·31-s − 976.·33-s + 78.8·35-s + 5.05e3·37-s + 1.45e3·39-s − 1.37e4·41-s − 1.72e4·43-s − 409.·45-s − 1.56e4·47-s − 1.47e4·49-s − 2.66e3·51-s + 3.07e4·53-s − 722.·55-s + ⋯ |
| L(s) = 1 | + 0.149·3-s + 0.0308·5-s + 0.353·7-s − 0.977·9-s − 1.04·11-s + 1.02·13-s + 0.00460·15-s − 0.961·17-s + 0.626·19-s + 0.0527·21-s − 1.15·23-s − 0.999·25-s − 0.295·27-s − 1.00·29-s + 0.179·31-s − 0.156·33-s + 0.0108·35-s + 0.606·37-s + 0.153·39-s − 1.28·41-s − 1.42·43-s − 0.0301·45-s − 1.03·47-s − 0.875·49-s − 0.143·51-s + 1.50·53-s − 0.0321·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - 961T \) |
| good | 3 | \( 1 - 2.32T + 243T^{2} \) |
| 5 | \( 1 - 1.72T + 3.12e3T^{2} \) |
| 7 | \( 1 - 45.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 419.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 626.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 985.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.57e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 5.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.37e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.54e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.81e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e5T + 8.58e9T^{2} \) |
| 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
−11.75073909333773348748495244021, −11.07284415967897646721452889191, −9.869704261358264300879232035648, −8.570888753761952908405719566025, −7.85125300701435861190225591754, −6.24924205496904580077646873964, −5.16950921299098673748245888664, −3.55053579800261356112642434512, −2.05378378433527642103298201338, 0,
2.05378378433527642103298201338, 3.55053579800261356112642434512, 5.16950921299098673748245888664, 6.24924205496904580077646873964, 7.85125300701435861190225591754, 8.570888753761952908405719566025, 9.869704261358264300879232035648, 11.07284415967897646721452889191, 11.75073909333773348748495244021
