| L(s) = 1 | − 20·3-s − 200·7-s + 55·9-s + 196·11-s − 360·13-s + 1.49e3·17-s + 3.18e3·19-s + 4.00e3·21-s − 1.56e3·23-s + 940·27-s − 3.92e3·29-s + 1.09e3·31-s − 3.92e3·33-s + 2.02e3·37-s + 7.20e3·39-s + 2.77e4·41-s + 3.00e3·43-s − 2.57e4·47-s − 2.65e3·49-s − 2.98e4·51-s − 2.69e4·53-s − 6.36e4·57-s − 1.19e4·59-s − 2.43e4·61-s − 1.10e4·63-s + 4.00e4·67-s + 3.12e4·69-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.54·7-s + 0.226·9-s + 0.488·11-s − 0.590·13-s + 1.25·17-s + 2.02·19-s + 1.97·21-s − 0.614·23-s + 0.248·27-s − 0.865·29-s + 0.204·31-s − 0.626·33-s + 0.242·37-s + 0.758·39-s + 2.57·41-s + 0.247·43-s − 1.70·47-s − 0.157·49-s − 1.60·51-s − 1.31·53-s − 2.59·57-s − 0.447·59-s − 0.839·61-s − 0.349·63-s + 1.09·67-s + 0.788·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 20 T + 115 p T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 200 T + 42650 T^{2} + 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 196 T + 181081 T^{2} - 196 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 360 T + 713290 T^{2} + 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1490 T + 2280355 T^{2} - 1490 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1560 T + 13472410 T^{2} + 1560 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3920 T + 44478298 T^{2} + 3920 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1096 T - 15343894 T^{2} - 1096 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2020 T + 130823790 T^{2} - 2020 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3000 T + 23431750 T^{2} - 3000 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 25760 T + 480952270 T^{2} + 25760 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 26980 T + 984299470 T^{2} + 26980 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11960 T + 1234152598 T^{2} + 11960 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 40060 T + 2949898505 T^{2} - 40060 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 87296 T + 5453356606 T^{2} - 87296 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 70290 T + 5306812435 T^{2} + 70290 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 92580 T + 9976491505 T^{2} + 92580 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 72810 T + 5926578523 T^{2} + 72810 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 126140 T + 13936294470 T^{2} + 126140 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07166772226555328314995621015, −9.663760769952137382378764424603, −9.505408286637826905360595067277, −9.326389073826344049558828735669, −8.143048867060283140636510069865, −7.925245792904650260681368811728, −7.26543367833479084050704529738, −6.89649637948758535004780645297, −6.15034666101399144060676425011, −6.07598229010620815568871447937, −5.36772197524814728456955164409, −5.24612615091120063156159555819, −4.27965027344745055652818042099, −3.74478506992973619234538897316, −2.98033014974405254573320178382, −2.83916462036254782026243317163, −1.50412646747505076647988632968, −0.999528630224857704090680346011, 0, 0,
0.999528630224857704090680346011, 1.50412646747505076647988632968, 2.83916462036254782026243317163, 2.98033014974405254573320178382, 3.74478506992973619234538897316, 4.27965027344745055652818042099, 5.24612615091120063156159555819, 5.36772197524814728456955164409, 6.07598229010620815568871447937, 6.15034666101399144060676425011, 6.89649637948758535004780645297, 7.26543367833479084050704529738, 7.925245792904650260681368811728, 8.143048867060283140636510069865, 9.326389073826344049558828735669, 9.505408286637826905360595067277, 9.663760769952137382378764424603, 10.07166772226555328314995621015
