| L(s) = 1 | + 8·5-s − 24·7-s + 2·11-s − 32·13-s + 112·17-s + 368·19-s + 92·23-s − 31·25-s − 336·29-s − 376·31-s − 192·35-s + 696·37-s − 312·41-s − 80·43-s − 228·47-s + 533·49-s − 304·53-s + 16·55-s + 680·59-s + 112·61-s − 256·65-s − 352·67-s − 3.63e3·71-s − 1.14e3·73-s − 48·77-s + 1.36e3·79-s − 782·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 1.29·7-s + 0.0548·11-s − 0.682·13-s + 1.59·17-s + 4.44·19-s + 0.834·23-s − 0.247·25-s − 2.15·29-s − 2.17·31-s − 0.927·35-s + 3.09·37-s − 1.18·41-s − 0.283·43-s − 0.707·47-s + 1.55·49-s − 0.787·53-s + 0.0392·55-s + 1.50·59-s + 0.235·61-s − 0.488·65-s − 0.641·67-s − 6.07·71-s − 1.84·73-s − 0.0710·77-s + 1.93·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2962806442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2962806442\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T + 19 p T^{2} + 2248 T^{3} - 22664 T^{4} + 2248 p^{3} T^{5} + 19 p^{7} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 24 T + 43 T^{2} - 3672 T^{3} - 31128 T^{4} - 3672 p^{3} T^{5} + 43 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 1330 T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 32 T + 1126 T^{2} - 143872 T^{3} - 7066133 T^{4} - 143872 p^{3} T^{5} + 1126 p^{6} T^{6} + 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 56 T + 5858 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 184 T + 20994 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 p T - 13234 T^{2} + 10544 p T^{3} + 219785827 T^{4} + 10544 p^{4} T^{5} - 13234 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 336 T + 37082 T^{2} + 9084096 T^{3} + 2399518731 T^{4} + 9084096 p^{3} T^{5} + 37082 p^{6} T^{6} + 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 376 T + 53875 T^{2} + 10497544 T^{3} + 2557188904 T^{4} + 10497544 p^{3} T^{5} + 53875 p^{6} T^{6} + 376 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 348 T + 126830 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 312 T - 35134 T^{2} - 1673568 T^{3} + 6091270419 T^{4} - 1673568 p^{3} T^{5} - 35134 p^{6} T^{6} + 312 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 80 T - 96002 T^{2} - 4528960 T^{3} + 3655961755 T^{4} - 4528960 p^{3} T^{5} - 96002 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 228 T - 149650 T^{2} - 1370736 T^{3} + 24479919795 T^{4} - 1370736 p^{3} T^{5} - 149650 p^{6} T^{6} + 228 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 152 T + 253337 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 680 T + 107114 T^{2} + 37720960 T^{3} - 10919641445 T^{4} + 37720960 p^{3} T^{5} + 107114 p^{6} T^{6} - 680 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 112 T - 140426 T^{2} + 33711104 T^{3} - 30967803125 T^{4} + 33711104 p^{3} T^{5} - 140426 p^{6} T^{6} - 112 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 352 T - 165266 T^{2} - 109949312 T^{3} - 44104832021 T^{4} - 109949312 p^{3} T^{5} - 165266 p^{6} T^{6} + 352 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1816 T + 1497518 T^{2} + 1816 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 287 T + p^{3} T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1360 T + 705250 T^{2} - 215249920 T^{3} + 154524532579 T^{4} - 215249920 p^{3} T^{5} + 705250 p^{6} T^{6} - 1360 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 782 T - 380803 T^{2} - 118275154 T^{3} + 308111028172 T^{4} - 118275154 p^{3} T^{5} - 380803 p^{6} T^{6} + 782 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 240 T + 1328110 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 338 T - 1720655 T^{2} - 3228914 T^{3} + 2429614921060 T^{4} - 3228914 p^{3} T^{5} - 1720655 p^{6} T^{6} - 338 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29831179919883849203230939977, −7.00016053671822359861449893149, −6.77542452850580058567411460570, −6.57463453410536706663789955438, −5.99501233571090602245326591081, −5.89577746237190044180863581545, −5.71481174197785179481437762847, −5.46935189126835123455420737121, −5.41120232621323045105962354891, −5.11012924042025451958135629171, −5.03745689349278820842144531442, −4.21878586349458622119873033549, −4.15363750772972590400842731673, −4.04350433246387124542201485421, −3.46607680167183040452817112205, −3.17882134052571890939206811343, −3.01292408779753122208755119192, −2.85579461227735792296342099087, −2.85287475066284893017967416707, −1.93949131146228139289280614569, −1.75941171531223392201104105054, −1.27965750474988099893347685210, −1.19466006460946623437018804204, −0.71504875458217010143526142182, −0.07267219774479793732515836395,
0.07267219774479793732515836395, 0.71504875458217010143526142182, 1.19466006460946623437018804204, 1.27965750474988099893347685210, 1.75941171531223392201104105054, 1.93949131146228139289280614569, 2.85287475066284893017967416707, 2.85579461227735792296342099087, 3.01292408779753122208755119192, 3.17882134052571890939206811343, 3.46607680167183040452817112205, 4.04350433246387124542201485421, 4.15363750772972590400842731673, 4.21878586349458622119873033549, 5.03745689349278820842144531442, 5.11012924042025451958135629171, 5.41120232621323045105962354891, 5.46935189126835123455420737121, 5.71481174197785179481437762847, 5.89577746237190044180863581545, 5.99501233571090602245326591081, 6.57463453410536706663789955438, 6.77542452850580058567411460570, 7.00016053671822359861449893149, 7.29831179919883849203230939977
