| L(s) = 1 | − 36·5-s + 108·9-s − 36·11-s − 516·23-s − 2.99e3·25-s − 2.75e3·31-s + 5.29e3·37-s − 3.88e3·45-s − 420·47-s + 6.18e3·49-s + 3.54e3·53-s + 1.29e3·55-s + 1.66e4·59-s + 3.65e3·67-s + 1.32e4·71-s + 7.29e3·81-s + 1.55e4·89-s + 7.62e3·97-s − 3.88e3·99-s − 2.45e4·103-s − 7.36e4·113-s + 1.85e4·115-s + 6.11e3·121-s + 1.29e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 4/3·9-s − 0.297·11-s − 0.975·23-s − 4.78·25-s − 2.86·31-s + 3.86·37-s − 1.91·45-s − 0.190·47-s + 2.57·49-s + 1.26·53-s + 0.428·55-s + 4.77·59-s + 0.814·67-s + 2.62·71-s + 10/9·81-s + 1.96·89-s + 0.810·97-s − 0.396·99-s − 2.31·103-s − 5.76·113-s + 1.40·115-s + 0.417·121-s + 8.31·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(18.53743335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.53743335\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 11 | \( 1 + 36 T - 4816 T^{2} + 101940 p T^{3} + 120330 p^{3} T^{4} + 101940 p^{5} T^{5} - 4816 p^{8} T^{6} + 36 p^{12} T^{7} + p^{16} T^{8} \) |
| good | 5 | \( ( 1 + 18 T + 1982 T^{2} + 30438 T^{3} + 1746834 T^{4} + 30438 p^{4} T^{5} + 1982 p^{8} T^{6} + 18 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 7 | \( 1 - 884 p T^{2} + 11678572 T^{4} - 930952796 p T^{6} + 1490364730198 T^{8} - 930952796 p^{9} T^{10} + 11678572 p^{16} T^{12} - 884 p^{25} T^{14} + p^{32} T^{16} \) |
| 13 | \( 1 - 162008 T^{2} + 12590117968 T^{4} - 619436460996728 T^{6} + 21062441542359634558 T^{8} - 619436460996728 p^{8} T^{10} + 12590117968 p^{16} T^{12} - 162008 p^{24} T^{14} + p^{32} T^{16} \) |
| 17 | \( 1 - 341468 T^{2} + 59012100712 T^{4} - 6828905158505300 T^{6} + \)\(62\!\cdots\!46\)\( T^{8} - 6828905158505300 p^{8} T^{10} + 59012100712 p^{16} T^{12} - 341468 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( 1 - 30284 p T^{2} + 152808356152 T^{4} - 26685780964652108 T^{6} + \)\(37\!\cdots\!98\)\( T^{8} - 26685780964652108 p^{8} T^{10} + 152808356152 p^{16} T^{12} - 30284 p^{25} T^{14} + p^{32} T^{16} \) |
| 23 | \( ( 1 + 258 T + 660806 T^{2} + 61700622 T^{3} + 215285764002 T^{4} + 61700622 p^{4} T^{5} + 660806 p^{8} T^{6} + 258 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 29 | \( 1 - 2005364 T^{2} + 2077725355336 T^{4} - 1841644838469189788 T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - 1841644838469189788 p^{8} T^{10} + 2077725355336 p^{16} T^{12} - 2005364 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 1376 T + 3679672 T^{2} + 3640691072 T^{3} + 5096962849966 T^{4} + 3640691072 p^{4} T^{5} + 3679672 p^{8} T^{6} + 1376 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2648 T + 9245032 T^{2} - 15286625576 T^{3} + 27654983547214 T^{4} - 15286625576 p^{4} T^{5} + 9245032 p^{8} T^{6} - 2648 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 3972980 T^{2} + 15101282902792 T^{4} - 53462970682481231900 T^{6} + \)\(21\!\cdots\!90\)\( T^{8} - 53462970682481231900 p^{8} T^{10} + 15101282902792 p^{16} T^{12} - 3972980 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( 1 - 18385892 T^{2} + 170175980842552 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(40\!\cdots\!14\)\( T^{8} - \)\(10\!\cdots\!76\)\( p^{8} T^{10} + 170175980842552 p^{16} T^{12} - 18385892 p^{24} T^{14} + p^{32} T^{16} \) |
| 47 | \( ( 1 + 210 T + 12240722 T^{2} + 2782593654 T^{3} + 84242391941250 T^{4} + 2782593654 p^{4} T^{5} + 12240722 p^{8} T^{6} + 210 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 1770 T + 319702 p T^{2} - 10757127678 T^{3} + 127498934985138 T^{4} - 10757127678 p^{4} T^{5} + 319702 p^{9} T^{6} - 1770 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8316 T + 56229104 T^{2} - 271941082596 T^{3} + 17443356249882 p T^{4} - 271941082596 p^{4} T^{5} + 56229104 p^{8} T^{6} - 8316 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 61 | \( 1 - 45292376 T^{2} + 1317023575398064 T^{4} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(43\!\cdots\!50\)\( T^{8} - \)\(27\!\cdots\!16\)\( p^{8} T^{10} + 1317023575398064 p^{16} T^{12} - 45292376 p^{24} T^{14} + p^{32} T^{16} \) |
| 67 | \( ( 1 - 1828 T + 26966776 T^{2} + 111245917268 T^{3} + 82170182924782 T^{4} + 111245917268 p^{4} T^{5} + 26966776 p^{8} T^{6} - 1828 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 6606 T + 54779210 T^{2} - 58180973226 T^{3} + 636995684602866 T^{4} - 58180973226 p^{4} T^{5} + 54779210 p^{8} T^{6} - 6606 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 73 | \( 1 - 165039416 T^{2} + 13195111482415036 T^{4} - \)\(65\!\cdots\!96\)\( T^{6} + \)\(22\!\cdots\!74\)\( T^{8} - \)\(65\!\cdots\!96\)\( p^{8} T^{10} + 13195111482415036 p^{16} T^{12} - 165039416 p^{24} T^{14} + p^{32} T^{16} \) |
| 79 | \( 1 + 7078036 T^{2} + 3829765928741164 T^{4} + \)\(41\!\cdots\!64\)\( T^{6} + \)\(77\!\cdots\!14\)\( T^{8} + \)\(41\!\cdots\!64\)\( p^{8} T^{10} + 3829765928741164 p^{16} T^{12} + 7078036 p^{24} T^{14} + p^{32} T^{16} \) |
| 83 | \( 1 - 135574280 T^{2} + 10334922471866332 T^{4} - \)\(61\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!50\)\( T^{8} - \)\(61\!\cdots\!96\)\( p^{8} T^{10} + 10334922471866332 p^{16} T^{12} - 135574280 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 - 7764 T + 181541324 T^{2} - 755869455948 T^{3} + 13637128342858662 T^{4} - 755869455948 p^{4} T^{5} + 181541324 p^{8} T^{6} - 7764 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 3812 T + 232957276 T^{2} - 518258717228 T^{3} + 25105593700183222 T^{4} - 518258717228 p^{4} T^{5} + 232957276 p^{8} T^{6} - 3812 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−4.10189177503878156553185677919, −4.10011072195342540785718322307, −3.84923225567264053284948838211, −3.75398995812854343214628591111, −3.59746752929155843048957921255, −3.51405911170075611558784027875, −3.30448146915026431661192198669, −3.09663630008384763166420782819, −3.03054694536655040409033361901, −2.56114477933822682782015265531, −2.43355163907506626760155906545, −2.42704077265723914757553172484, −2.24342288004130888900179307542, −2.11336299606985553517634299869, −2.06207166414384513715191832869, −1.74600445000380665995600877249, −1.64132692531162613739628390481, −1.54571429052643480669467400780, −1.28439546739796148458096700079, −0.886817712932012517965164771715, −0.61408682942961136964102880626, −0.60153114523430196745108686380, −0.47929545171875344011469292576, −0.46252358501638968978936032141, −0.32297364978766454038814305408,
0.32297364978766454038814305408, 0.46252358501638968978936032141, 0.47929545171875344011469292576, 0.60153114523430196745108686380, 0.61408682942961136964102880626, 0.886817712932012517965164771715, 1.28439546739796148458096700079, 1.54571429052643480669467400780, 1.64132692531162613739628390481, 1.74600445000380665995600877249, 2.06207166414384513715191832869, 2.11336299606985553517634299869, 2.24342288004130888900179307542, 2.42704077265723914757553172484, 2.43355163907506626760155906545, 2.56114477933822682782015265531, 3.03054694536655040409033361901, 3.09663630008384763166420782819, 3.30448146915026431661192198669, 3.51405911170075611558784027875, 3.59746752929155843048957921255, 3.75398995812854343214628591111, 3.84923225567264053284948838211, 4.10011072195342540785718322307, 4.10189177503878156553185677919
Plot not available for L-functions of degree greater than 10.
