| L(s) = 1 | − 40·5-s + 56·13-s − 744·17-s − 1.40e3·25-s − 8·29-s + 3.67e3·37-s + 152·41-s + 3.33e3·49-s + 1.16e4·53-s + 1.66e4·61-s − 2.24e3·65-s − 2.71e4·73-s + 2.97e4·85-s − 2.06e4·89-s + 8.20e3·97-s − 1.93e4·101-s − 2.21e4·109-s + 2.51e4·113-s + 3.60e4·121-s + 9.92e4·125-s + 127-s + 131-s + 137-s + 139-s + 320·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 0.331·13-s − 2.57·17-s − 2.24·25-s − 0.00951·29-s + 2.68·37-s + 0.0904·41-s + 1.38·49-s + 4.14·53-s + 4.47·61-s − 0.530·65-s − 5.10·73-s + 4.11·85-s − 2.60·89-s + 0.871·97-s − 1.89·101-s − 1.86·109-s + 1.96·113-s + 2.46·121-s + 6.35·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.0152·145-s + 4.50e−5·149-s + 4.38e−5·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6846276193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6846276193\) |
| \(L(3)\) |
|
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 | $D_{4}$ | \( ( 1 + 4 p T + 1302 T^{2} + 4 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 68 p^{2} T^{2} + 5012358 T^{4} - 68 p^{10} T^{6} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36068 T^{2} + 652185990 T^{4} - 36068 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 28 T + 1830 T^{2} - 28 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 372 T + 178406 T^{2} + 372 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 342628 T^{2} + 63111128070 T^{4} - 342628 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 618500 T^{2} + 250665008262 T^{4} - 618500 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 75894 T^{2} + 4 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3094148 T^{2} + 4071331901190 T^{4} - 3094148 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 1836 T + 3171014 T^{2} - 1836 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 76 T - 498906 T^{2} - 76 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1399268 T^{2} + 21307575337350 T^{4} - 1399268 p^{8} T^{6} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9607940 T^{2} + 66483303721734 T^{4} - 9607940 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 5820 T + 24143030 T^{2} - 5820 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 16992412 T^{2} + 193877183283078 T^{4} + 16992412 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 8332 T + 44051910 T^{2} - 8332 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 17541092 T^{2} - 102797034147834 T^{4} - 17541092 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 42435332 T^{2} + 1131568734506886 T^{4} - 42435332 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 13596 T + 102858758 T^{2} + 13596 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 148035716 T^{2} + 8503008803186694 T^{4} - 148035716 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 143213156 T^{2} + 9600231837327366 T^{4} - 143213156 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 10308 T + 58505030 T^{2} + 10308 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 4100 T + 47714310 T^{2} - 4100 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−7.15125741232837762854853451531, −6.99761422892944577467805558825, −6.85598889036051160553957109973, −6.27123532220240911874653322139, −6.09911666537232480151076306055, −5.83594122314255884907610876548, −5.80140683469341042040551349015, −5.48604413725437441667733745986, −5.19674286929057551230278910152, −4.50537107458518072447567950594, −4.50044574831966267108956341532, −4.43352642450893944415625853773, −3.88564184122200644786452244927, −3.87323702216220511950270748978, −3.85732745218358276032430365286, −3.43865260230357030566654482346, −2.73072869899987064977522203394, −2.47051493496187137449902788422, −2.46500089726581121327335745771, −2.16122383619991102448299004937, −1.70805068915916957357093777114, −1.06305148300022755324963466954, −0.993972179951929396163434862518, −0.37255233491219070295172179401, −0.18136260211672025118581299972,
0.18136260211672025118581299972, 0.37255233491219070295172179401, 0.993972179951929396163434862518, 1.06305148300022755324963466954, 1.70805068915916957357093777114, 2.16122383619991102448299004937, 2.46500089726581121327335745771, 2.47051493496187137449902788422, 2.73072869899987064977522203394, 3.43865260230357030566654482346, 3.85732745218358276032430365286, 3.87323702216220511950270748978, 3.88564184122200644786452244927, 4.43352642450893944415625853773, 4.50044574831966267108956341532, 4.50537107458518072447567950594, 5.19674286929057551230278910152, 5.48604413725437441667733745986, 5.80140683469341042040551349015, 5.83594122314255884907610876548, 6.09911666537232480151076306055, 6.27123532220240911874653322139, 6.85598889036051160553957109973, 6.99761422892944577467805558825, 7.15125741232837762854853451531
