| L(s) = 1 | + 8·5-s + 8·13-s + 32·25-s + 8·29-s + 40·37-s + 24·49-s − 8·53-s + 40·61-s + 64·65-s + 64·97-s − 40·101-s − 40·109-s − 16·113-s + 88·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 2.21·13-s + 32/5·25-s + 1.48·29-s + 6.57·37-s + 24/7·49-s − 1.09·53-s + 5.12·61-s + 7.93·65-s + 6.49·97-s − 3.98·101-s − 3.83·109-s − 1.50·113-s + 7.87·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(186.3933070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(186.3933070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 460 T^{4} + 82054 T^{8} - 460 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 4 T + 8 T^{2} - 44 T^{3} + 238 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 - 12 T^{4} - 247354 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 12 T^{2} + 1062 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T + 8 T^{2} + 20 T^{3} - 1106 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 11662 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 6068 T^{4} + 15827590 T^{8} + 6068 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + 8 T^{2} + 76 T^{3} - 434 T^{4} + 76 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 12084 T^{4} + 60324614 T^{8} + 12084 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 11054 T^{4} - 1500 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 116 T^{4} - 21877946 T^{8} + 116 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 76 T^{2} + 9958 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 3468 T^{4} - 36032314 T^{8} - 3468 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 332 T^{2} + 43270 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.41696128276895110391714616667, −3.15851811930828321152686284016, −3.09504613752625643574316275575, −3.01981250196201068052736523674, −2.83204106152622946562057164408, −2.80322581609463217464152975720, −2.70563927603881940807571181723, −2.59569721007837986639274239468, −2.37775643305926930730479354415, −2.36113298531816451154477593836, −2.17941603945069960835437120383, −2.17490900523370273450167153554, −2.10405051288651657384971029825, −2.05144379600198659945138249423, −1.69905062935868416396924989287, −1.52665349528921096232533622917, −1.43709692688999567739477180668, −1.38471641292717738123055963878, −1.16615148623903779682047023868, −1.06969652722801201432537508578, −0.967463228902003845746388201196, −0.76025280047799587773064252079, −0.55605295444972323745065606137, −0.54063107941873493671479739961, −0.51622372883477867218206555790,
0.51622372883477867218206555790, 0.54063107941873493671479739961, 0.55605295444972323745065606137, 0.76025280047799587773064252079, 0.967463228902003845746388201196, 1.06969652722801201432537508578, 1.16615148623903779682047023868, 1.38471641292717738123055963878, 1.43709692688999567739477180668, 1.52665349528921096232533622917, 1.69905062935868416396924989287, 2.05144379600198659945138249423, 2.10405051288651657384971029825, 2.17490900523370273450167153554, 2.17941603945069960835437120383, 2.36113298531816451154477593836, 2.37775643305926930730479354415, 2.59569721007837986639274239468, 2.70563927603881940807571181723, 2.80322581609463217464152975720, 2.83204106152622946562057164408, 3.01981250196201068052736523674, 3.09504613752625643574316275575, 3.15851811930828321152686284016, 3.41696128276895110391714616667
Plot not available for L-functions of degree greater than 10.
