| L(s) = 1 | − 169.·7-s + 290.·11-s − 639.·13-s + 129.·17-s + 971.·19-s + 849.·23-s + 8.04e3·29-s + 5.30e3·31-s − 4.51e3·37-s − 9.06e3·41-s − 1.52e3·43-s − 1.90e4·47-s + 1.20e4·49-s + 3.65e4·53-s − 1.69e4·59-s + 2.45e4·61-s + 4.95e4·67-s − 4.51e4·71-s − 2.92e4·73-s − 4.92e4·77-s + 1.34e4·79-s − 8.19e4·83-s − 6.77e4·89-s + 1.08e5·91-s + 1.01e5·97-s − 1.87e5·101-s − 1.82e5·103-s + ⋯ |
| L(s) = 1 | − 1.30·7-s + 0.723·11-s − 1.05·13-s + 0.108·17-s + 0.617·19-s + 0.334·23-s + 1.77·29-s + 0.990·31-s − 0.542·37-s − 0.842·41-s − 0.125·43-s − 1.25·47-s + 0.714·49-s + 1.78·53-s − 0.634·59-s + 0.844·61-s + 1.34·67-s − 1.06·71-s − 0.641·73-s − 0.946·77-s + 0.242·79-s − 1.30·83-s − 0.906·89-s + 1.37·91-s + 1.09·97-s − 1.83·101-s − 1.69·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 169.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 639.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 129.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 971.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 849.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
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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
−9.059378074473401718522291901389, −8.171681309455497937424879392788, −6.93970589681501820419533169945, −6.62212394475155023752539374626, −5.47122229846635955138204961367, −4.47472395428627716795041254229, −3.35480864821603634137842663063, −2.60941988616092604642317340866, −1.12188294920369126566754634216, 0,
1.12188294920369126566754634216, 2.60941988616092604642317340866, 3.35480864821603634137842663063, 4.47472395428627716795041254229, 5.47122229846635955138204961367, 6.62212394475155023752539374626, 6.93970589681501820419533169945, 8.171681309455497937424879392788, 9.059378074473401718522291901389
