| L(s) = 1 | − 2·2-s − 5·4-s − 22·5-s + 12·8-s + 44·10-s + 62·11-s + 64·13-s − 11·16-s + 32·17-s − 162·19-s + 110·20-s − 124·22-s + 170·23-s + 115·25-s − 128·26-s − 128·29-s − 178·31-s + 122·32-s − 64·34-s − 350·37-s + 324·38-s − 264·40-s − 58·41-s + 756·43-s − 310·44-s − 340·46-s + 180·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 5/8·4-s − 1.96·5-s + 0.530·8-s + 1.39·10-s + 1.69·11-s + 1.36·13-s − 0.171·16-s + 0.456·17-s − 1.95·19-s + 1.22·20-s − 1.20·22-s + 1.54·23-s + 0.919·25-s − 0.965·26-s − 0.819·29-s − 1.03·31-s + 0.673·32-s − 0.322·34-s − 1.55·37-s + 1.38·38-s − 1.04·40-s − 0.220·41-s + 2.68·43-s − 1.06·44-s − 1.08·46-s + 0.558·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 9 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 22 T + 369 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 62 T + 3551 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 64 T + 4840 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 32 T + 9434 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 162 T + 19311 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 170 T + 24117 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 128 T + 52152 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 178 T + 39181 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 350 T + 129753 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 58 T + 128315 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 756 T + 288450 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 180 T + 104354 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1088 T + 592538 T^{2} - 1088 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 508 T + 174186 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 152 T + 459666 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 468 T + 88104 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 193629 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 788 T + 930382 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 476 T + 940570 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1492 T + 1690008 T^{2} + 1492 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 230 T - 8269 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 640 T + 1770946 T^{2} - 640 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.825598914480012675049488847398, −8.808765367792287276066199686761, −8.225301418840038481317918372500, −8.204868214447504008470033654239, −7.38208223884772472850539389718, −7.11735733582857931315557282028, −6.86058063298896491914217281851, −6.25286433625253564777466875682, −5.67431819133354653054082283609, −5.39215020820646574515783636867, −4.34574248222921015659893481514, −4.25101284896703264319687919152, −3.84166157028382341677910983629, −3.75639303810444641708480717361, −2.96470147417517952847425134780, −2.15682772641896408220025382330, −1.24380007062860037366932254682, −1.00912637869304958416781753381, 0, 0,
1.00912637869304958416781753381, 1.24380007062860037366932254682, 2.15682772641896408220025382330, 2.96470147417517952847425134780, 3.75639303810444641708480717361, 3.84166157028382341677910983629, 4.25101284896703264319687919152, 4.34574248222921015659893481514, 5.39215020820646574515783636867, 5.67431819133354653054082283609, 6.25286433625253564777466875682, 6.86058063298896491914217281851, 7.11735733582857931315557282028, 7.38208223884772472850539389718, 8.204868214447504008470033654239, 8.225301418840038481317918372500, 8.808765367792287276066199686761, 8.825598914480012675049488847398
