| L(s) = 1 | + 6.07e3·7-s + 4.85e5·25-s + 2.49e6·37-s + 1.83e7·43-s + 1.14e7·49-s + 1.23e7·67-s + 5.32e7·79-s + 7.65e8·109-s − 5.63e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.27e9·169-s + 173-s + 2.94e9·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2.53·7-s + 1.24·25-s + 1.33·37-s + 5.36·43-s + 1.98·49-s + 0.611·67-s + 1.36·79-s + 5.42·109-s − 2.63·121-s + 6.46·169-s + 3.14·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(21.09924629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.09924629\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 62 p^{2} T + 3378 p^{4} T^{2} - 62 p^{10} T^{3} + p^{16} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 9706 p^{2} T^{2} + 16250483082 p T^{4} - 9706 p^{18} T^{6} + p^{32} T^{8} )^{2} \) |
| 11 | \( ( 1 + 281948794 T^{2} + 7352143829555286 p T^{4} + 281948794 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2635070548 T^{2} + 2979830483082530598 T^{4} - 2635070548 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 17 | \( ( 1 - 10174275994 T^{2} + \)\(10\!\cdots\!06\)\( T^{4} - 10174275994 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 19 | \( ( 1 - 13006623328 T^{2} + \)\(53\!\cdots\!98\)\( T^{4} - 13006623328 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 23 | \( ( 1 + 181727772634 T^{2} + \)\(18\!\cdots\!26\)\( T^{4} + 181727772634 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 29 | \( ( 1 + 829962446884 T^{2} + \)\(54\!\cdots\!46\)\( T^{4} + 829962446884 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3342446785828 T^{2} + \)\(42\!\cdots\!18\)\( T^{4} - 3342446785828 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 37 | \( ( 1 - 623774 T + 6944563834626 T^{2} - 623774 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 41 | \( ( 1 - 12738079335034 T^{2} + \)\(13\!\cdots\!46\)\( T^{4} - 12738079335034 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4585058 T + 15013574669418 T^{2} - 4585058 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 47 | \( ( 1 - 89095935957604 T^{2} + \)\(31\!\cdots\!06\)\( T^{4} - 89095935957604 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 53 | \( ( 1 + 178290223511044 T^{2} + \)\(15\!\cdots\!66\)\( T^{4} + 178290223511044 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 59 | \( ( 1 - 180211322565604 T^{2} + \)\(37\!\cdots\!46\)\( T^{4} - 180211322565604 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 61 | \( ( 1 + 175072976949932 T^{2} + \)\(77\!\cdots\!38\)\( T^{4} + 175072976949932 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 67 | \( ( 1 - 3080062 T + 777400454319978 T^{2} - 3080062 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 71 | \( ( 1 + 5928934903574 p T^{2} + \)\(62\!\cdots\!86\)\( T^{4} + 5928934903574 p^{17} T^{6} + p^{32} T^{8} )^{2} \) |
| 73 | \( ( 1 - 1012625463466564 T^{2} + \)\(13\!\cdots\!46\)\( T^{4} - 1012625463466564 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 79 | \( ( 1 - 13309474 T + 2879736805586226 T^{2} - 13309474 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 83 | \( ( 1 - 5030272623099364 T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - 5030272623099364 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 89 | \( ( 1 - 14262440598788314 T^{2} + \)\(10\!\cdots\!86\)\( p^{2} T^{4} - 14262440598788314 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 97 | \( ( 1 - 18239650877341828 T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - 18239650877341828 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
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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
−3.93457728112961058103138872804, −3.81419561873638821797183211335, −3.73160425112948031818987979170, −3.52419774869759986091098373246, −3.27404941966337661007364280617, −3.03948147999387760457402739343, −2.98882112626654435550493347761, −2.87552714135133931439123944368, −2.84913107140573108905384344038, −2.54016045849768746117732967692, −2.21963493415184230370301436523, −2.15277758201546996670079918881, −2.01412476921096773715522747760, −1.95933975073687462789154943741, −1.87809751614725136718164038542, −1.69178594369843542944134638230, −1.42319493034848362426543821497, −1.14806886047034354907739673327, −1.06411278524031688825679864643, −0.887704086759714670512053231976, −0.797431757855628430556781859612, −0.59624914969025484322418053346, −0.58885893747565996288010273612, −0.53256416305235541249355354379, −0.10505375140536629555119970275,
0.10505375140536629555119970275, 0.53256416305235541249355354379, 0.58885893747565996288010273612, 0.59624914969025484322418053346, 0.797431757855628430556781859612, 0.887704086759714670512053231976, 1.06411278524031688825679864643, 1.14806886047034354907739673327, 1.42319493034848362426543821497, 1.69178594369843542944134638230, 1.87809751614725136718164038542, 1.95933975073687462789154943741, 2.01412476921096773715522747760, 2.15277758201546996670079918881, 2.21963493415184230370301436523, 2.54016045849768746117732967692, 2.84913107140573108905384344038, 2.87552714135133931439123944368, 2.98882112626654435550493347761, 3.03948147999387760457402739343, 3.27404941966337661007364280617, 3.52419774869759986091098373246, 3.73160425112948031818987979170, 3.81419561873638821797183211335, 3.93457728112961058103138872804
Plot not available for L-functions of degree greater than 10.
