| L(s) = 1 | − 3.66e4·9-s + 1.32e6·13-s + 1.92e8·25-s − 2.09e7·37-s + 8.54e9·49-s + 5.52e9·61-s + 2.08e10·73-s − 1.80e10·81-s + 3.17e11·97-s + 8.12e11·109-s − 4.84e10·117-s − 3.24e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.13e11·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 0.207·9-s + 0.986·13-s + 3.94·25-s − 0.0496·37-s + 4.31·49-s + 0.837·61-s + 1.17·73-s − 0.574·81-s + 3.75·97-s + 5.05·109-s − 0.204·117-s − 0.0113·121-s + 0.453·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(47.72077899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(47.72077899\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4076 p^{2} T^{2} + 983986 p^{9} T^{4} + 4076 p^{24} T^{6} + p^{44} T^{8} \) |
| good | 5 | \( ( 1 - 19240324 p T^{2} + 55948037859246 p^{3} T^{4} - 19240324 p^{23} T^{6} + p^{44} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4270393636 T^{2} + 240056316670863174 p^{2} T^{4} - 4270393636 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 11 | \( ( 1 + 1622257580 T^{2} - \)\(96\!\cdots\!82\)\( T^{4} + 1622257580 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 13 | \( ( 1 - 330284 T + 69401776974 T^{2} - 330284 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 17 | \( ( 1 - 2827965939908 T^{2} + \)\(12\!\cdots\!50\)\( T^{4} - 2827965939908 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 19 | \( ( 1 - 223297938605812 T^{2} + \)\(39\!\cdots\!34\)\( T^{4} - 223297938605812 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 23 | \( ( 1 + 3430377369932444 T^{2} + \)\(47\!\cdots\!58\)\( T^{4} + 3430377369932444 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 29 | \( ( 1 - 12424717624883636 T^{2} + \)\(94\!\cdots\!06\)\( T^{4} - 12424717624883636 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 31 | \( ( 1 - 77078836605197188 T^{2} + \)\(27\!\cdots\!34\)\( T^{4} - 77078836605197188 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 37 | \( ( 1 + 5240516 T + 347337308980982526 T^{2} + 5240516 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2012267397302937188 T^{2} + \)\(16\!\cdots\!74\)\( T^{4} - 2012267397302937188 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 43 | \( ( 1 + 272720904026200364 T^{2} + \)\(15\!\cdots\!58\)\( T^{4} + 272720904026200364 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 47 | \( ( 1 - 582572705607044548 T^{2} + \)\(47\!\cdots\!94\)\( T^{4} - 582572705607044548 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16350974247114542804 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - 16350974247114542804 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 59 | \( ( 1 + 24480677761455243500 T^{2} - \)\(32\!\cdots\!62\)\( T^{4} + 24480677761455243500 p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 61 | \( ( 1 - 1380920140 T + 87298836341980577838 T^{2} - 1380920140 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 67 | \( ( 1 - \)\(25\!\cdots\!84\)\( T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - \)\(25\!\cdots\!84\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 71 | \( ( 1 + \)\(53\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!78\)\( T^{4} + \)\(53\!\cdots\!80\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 73 | \( ( 1 - 5204156468 T + \)\(43\!\cdots\!54\)\( T^{2} - 5204156468 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
| 79 | \( ( 1 - \)\(56\!\cdots\!32\)\( T^{2} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(56\!\cdots\!32\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 83 | \( ( 1 + \)\(22\!\cdots\!72\)\( T^{2} + \)\(39\!\cdots\!30\)\( T^{4} + \)\(22\!\cdots\!72\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 89 | \( ( 1 - \)\(63\!\cdots\!92\)\( T^{2} + \)\(24\!\cdots\!14\)\( T^{4} - \)\(63\!\cdots\!92\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) |
| 97 | \( ( 1 - 79479852740 T + \)\(12\!\cdots\!06\)\( T^{2} - 79479852740 p^{11} T^{3} + p^{22} T^{4} )^{4} \) |
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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
−4.91704109675118613682908859112, −4.80073542070388412114044380235, −4.60534391079795174765629254774, −4.56113851549167819330965219211, −4.16823915828791561489321812012, −4.14187716222742053703090312925, −3.99670517354088437699266979784, −3.51090531732624689572949147133, −3.37090286037508891441815203531, −3.30674307157151059710889262254, −3.06252732223843498804925078812, −3.03673293591491900749361972280, −2.67598085711123376141320402436, −2.36820892710093593805734067301, −2.28165575106427054732193420243, −2.11977590115148572207695048080, −1.75158476297717051988635366531, −1.72240073690486532060917315931, −1.39588479722256658944978826497, −1.00139472546401758113006719693, −0.901696032565655310542807251447, −0.71087811581451425662508948967, −0.57399676893809699367805656573, −0.56186939161752291434893142312, −0.38821487261995345634461241761,
0.38821487261995345634461241761, 0.56186939161752291434893142312, 0.57399676893809699367805656573, 0.71087811581451425662508948967, 0.901696032565655310542807251447, 1.00139472546401758113006719693, 1.39588479722256658944978826497, 1.72240073690486532060917315931, 1.75158476297717051988635366531, 2.11977590115148572207695048080, 2.28165575106427054732193420243, 2.36820892710093593805734067301, 2.67598085711123376141320402436, 3.03673293591491900749361972280, 3.06252732223843498804925078812, 3.30674307157151059710889262254, 3.37090286037508891441815203531, 3.51090531732624689572949147133, 3.99670517354088437699266979784, 4.14187716222742053703090312925, 4.16823915828791561489321812012, 4.56113851549167819330965219211, 4.60534391079795174765629254774, 4.80073542070388412114044380235, 4.91704109675118613682908859112
Plot not available for L-functions of degree greater than 10.
