| L(s) = 1 | − 32·4-s + 296·13-s + 640·16-s − 80·19-s + 1.76e3·25-s − 1.02e3·31-s + 904·37-s − 3.22e3·43-s + 1.37e3·49-s − 9.47e3·52-s + 128·61-s − 1.02e4·64-s − 1.01e4·67-s + 1.06e4·73-s + 2.56e3·76-s − 4.07e3·79-s + 3.56e4·97-s − 5.65e4·100-s − 3.38e4·103-s − 3.39e4·109-s + 8.56e4·121-s + 3.27e4·124-s + 127-s + 131-s + 137-s + 139-s − 2.89e4·148-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.75·13-s + 5/2·16-s − 0.221·19-s + 2.82·25-s − 1.06·31-s + 0.660·37-s − 1.74·43-s + 4/7·49-s − 3.50·52-s + 0.0343·61-s − 5/2·64-s − 2.26·67-s + 1.99·73-s + 0.443·76-s − 0.652·79-s + 3.78·97-s − 5.65·100-s − 3.18·103-s − 2.86·109-s + 5.84·121-s + 2.13·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 1.32·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.02662225220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02662225220\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| good | 5 | \( 1 - 1768 T^{2} + 2529134 T^{4} - 2252426112 T^{6} + 1676444306819 T^{8} - 2252426112 p^{8} T^{10} + 2529134 p^{16} T^{12} - 1768 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 7784 p T^{2} + 3325121246 T^{4} - 7244883084096 p T^{6} + 1353610316397331667 T^{8} - 7244883084096 p^{9} T^{10} + 3325121246 p^{16} T^{12} - 7784 p^{25} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 - 148 T + 39212 T^{2} - 9739428 T^{3} + 1738923098 T^{4} - 9739428 p^{4} T^{5} + 39212 p^{8} T^{6} - 148 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 625408 T^{2} + 174522303236 T^{4} - 28359358654324992 T^{6} + \)\(29\!\cdots\!42\)\( T^{8} - 28359358654324992 p^{8} T^{10} + 174522303236 p^{16} T^{12} - 625408 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 + 40 T + 17722 p T^{2} - 3495296 T^{3} + 53520965731 T^{4} - 3495296 p^{4} T^{5} + 17722 p^{9} T^{6} + 40 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1907944 T^{2} + 1664585572430 T^{4} - 869099980712802816 T^{6} + \)\(29\!\cdots\!87\)\( T^{8} - 869099980712802816 p^{8} T^{10} + 1664585572430 p^{16} T^{12} - 1907944 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 3864160 T^{2} + 6974374989956 T^{4} - 7963063098784827168 T^{6} + \)\(65\!\cdots\!58\)\( T^{8} - 7963063098784827168 p^{8} T^{10} + 6974374989956 p^{16} T^{12} - 3864160 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 512 T + 3209846 T^{2} + 1353628032 T^{3} + 4244878530971 T^{4} + 1353628032 p^{4} T^{5} + 3209846 p^{8} T^{6} + 512 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 452 T + 5739634 T^{2} - 2195537312 T^{3} + 15174196511899 T^{4} - 2195537312 p^{4} T^{5} + 5739634 p^{8} T^{6} - 452 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 17121992 T^{2} + 134745260445358 T^{4} - \)\(65\!\cdots\!84\)\( T^{6} + \)\(21\!\cdots\!03\)\( T^{8} - \)\(65\!\cdots\!84\)\( p^{8} T^{10} + 134745260445358 p^{16} T^{12} - 17121992 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 1612 T + 4253740 T^{2} + 8380356028 T^{3} + 25490935217626 T^{4} + 8380356028 p^{4} T^{5} + 4253740 p^{8} T^{6} + 1612 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 14023712 T^{2} + 156295616601604 T^{4} - \)\(10\!\cdots\!16\)\( T^{6} + \)\(63\!\cdots\!90\)\( T^{8} - \)\(10\!\cdots\!16\)\( p^{8} T^{10} + 156295616601604 p^{16} T^{12} - 14023712 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 34292168 T^{2} + 668473835183164 T^{4} - \)\(85\!\cdots\!20\)\( T^{6} + \)\(79\!\cdots\!90\)\( T^{8} - \)\(85\!\cdots\!20\)\( p^{8} T^{10} + 668473835183164 p^{16} T^{12} - 34292168 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 24661936 T^{2} + 172139831207780 T^{4} - \)\(32\!\cdots\!24\)\( T^{6} + \)\(73\!\cdots\!22\)\( T^{8} - \)\(32\!\cdots\!24\)\( p^{8} T^{10} + 172139831207780 p^{16} T^{12} - 24661936 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 64 T + 16531340 T^{2} + 65141281344 T^{3} + 111569841434726 T^{4} + 65141281344 p^{4} T^{5} + 16531340 p^{8} T^{6} - 64 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 5076 T + 59033612 T^{2} + 198232866468 T^{3} + 1460747434921818 T^{4} + 198232866468 p^{4} T^{5} + 59033612 p^{8} T^{6} + 5076 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 151904440 T^{2} + 10938525571935806 T^{4} - \)\(48\!\cdots\!92\)\( T^{6} + \)\(14\!\cdots\!51\)\( T^{8} - \)\(48\!\cdots\!92\)\( p^{8} T^{10} + 10938525571935806 p^{16} T^{12} - 151904440 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 5316 T + 99071756 T^{2} - 395704493652 T^{3} + 3980454703922394 T^{4} - 395704493652 p^{4} T^{5} + 99071756 p^{8} T^{6} - 5316 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2036 T + 109661996 T^{2} + 211792477380 T^{3} + 5959931103886106 T^{4} + 211792477380 p^{4} T^{5} + 109661996 p^{8} T^{6} + 2036 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 164741488 T^{2} + 16960421186852708 T^{4} - \)\(12\!\cdots\!72\)\( T^{6} + \)\(66\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!72\)\( p^{8} T^{10} + 16960421186852708 p^{16} T^{12} - 164741488 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 364989176 T^{2} + 64125406327249054 T^{4} - \)\(70\!\cdots\!32\)\( T^{6} + \)\(53\!\cdots\!07\)\( T^{8} - \)\(70\!\cdots\!32\)\( p^{8} T^{10} + 64125406327249054 p^{16} T^{12} - 364989176 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 - 17808 T + 401078828 T^{2} - 4611403938288 T^{3} + 55387080124031718 T^{4} - 4611403938288 p^{4} T^{5} + 401078828 p^{8} T^{6} - 17808 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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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
−4.34881730388686932047630722916, −4.29966590019293263704582730153, −4.13636867695753877131219151054, −3.74212359793793460041282192507, −3.71591298352670983082474059835, −3.63502655655678901055175483119, −3.41443271660039333620621557374, −3.40177278654584788309302066770, −3.16189182214235118765941646650, −3.10856963233416816976249377835, −2.76167326486954341224233009486, −2.74974396143754387814998713164, −2.48701624026654270958992322628, −2.21636027853596754229775978625, −2.10398796329188329171904854127, −1.78069874509891056203252195031, −1.64998887416714884736019908903, −1.47147771524509425892019343269, −1.30649766988200906457509731288, −0.978598553003939144593399100929, −0.951779355260507592884803160705, −0.78281982667495288420728384043, −0.49017266662901191726525545191, −0.40617980914339923484497117999, −0.01267088407016064652446326836,
0.01267088407016064652446326836, 0.40617980914339923484497117999, 0.49017266662901191726525545191, 0.78281982667495288420728384043, 0.951779355260507592884803160705, 0.978598553003939144593399100929, 1.30649766988200906457509731288, 1.47147771524509425892019343269, 1.64998887416714884736019908903, 1.78069874509891056203252195031, 2.10398796329188329171904854127, 2.21636027853596754229775978625, 2.48701624026654270958992322628, 2.74974396143754387814998713164, 2.76167326486954341224233009486, 3.10856963233416816976249377835, 3.16189182214235118765941646650, 3.40177278654584788309302066770, 3.41443271660039333620621557374, 3.63502655655678901055175483119, 3.71591298352670983082474059835, 3.74212359793793460041282192507, 4.13636867695753877131219151054, 4.29966590019293263704582730153, 4.34881730388686932047630722916
Plot not available for L-functions of degree greater than 10.
