| L(s) = 1 | − 512·4-s − 2.58e3·5-s + 3.38e3·9-s − 1.03e5·11-s + 1.96e5·16-s + 3.17e5·19-s + 1.32e6·20-s + 3.12e6·25-s + 6.66e6·29-s + 1.92e7·31-s − 1.73e6·36-s + 2.27e7·41-s + 5.30e7·44-s − 8.72e6·45-s + 2.67e7·49-s + 2.67e8·55-s + 2.54e8·59-s − 2.86e8·61-s − 6.71e7·64-s + 8.02e8·71-s − 1.62e8·76-s − 1.03e9·79-s − 5.07e8·80-s − 2.92e7·81-s − 2.76e8·89-s − 8.19e8·95-s − 3.50e8·99-s + ⋯ |
| L(s) = 1 | − 4-s − 1.84·5-s + 0.171·9-s − 2.13·11-s + 3/4·16-s + 0.558·19-s + 1.84·20-s + 1.60·25-s + 1.75·29-s + 3.74·31-s − 0.171·36-s + 1.25·41-s + 2.13·44-s − 0.317·45-s + 0.662·49-s + 3.93·55-s + 2.73·59-s − 2.65·61-s − 1/2·64-s + 3.74·71-s − 0.558·76-s − 2.98·79-s − 1.38·80-s − 0.0755·81-s − 0.466·89-s − 1.03·95-s − 0.366·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.099540459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099540459\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 516 p T + 5646 p^{4} T^{2} + 516 p^{10} T^{3} + p^{18} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 3380 T^{2} + 4519942 p^{2} T^{4} - 3380 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 26720900 T^{2} + 29838153055302 p^{2} T^{4} - 26720900 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 51816 T + 3027030246 T^{2} + 51816 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 22383928660 T^{2} + \)\(29\!\cdots\!58\)\( T^{4} - 22383928660 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 468221105220 T^{2} + \)\(82\!\cdots\!18\)\( T^{4} - 468221105220 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 158760 T + 642783370358 T^{2} - 158760 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 70724101980 p T^{2} + \)\(46\!\cdots\!38\)\( T^{4} - 70724101980 p^{19} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3334140 T + 25948591194238 T^{2} - 3334140 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 9623744 T + 71729043019326 T^{2} - 9623744 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 73233430078540 T^{2} + \)\(20\!\cdots\!58\)\( T^{4} + 73233430078540 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 11387124 T + 602060396517366 T^{2} - 11387124 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1938214042750100 T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - 1938214042750100 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3434853059674980 T^{2} + \)\(54\!\cdots\!78\)\( T^{4} - 3434853059674980 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3945786642536180 T^{2} + \)\(65\!\cdots\!78\)\( T^{4} - 3945786642536180 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 127330680 T + 20590638486439878 T^{2} - 127330680 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 143290916 T + 28088617153288446 T^{2} + 143290916 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 105856746688500020 T^{2} + \)\(42\!\cdots\!18\)\( T^{4} - 105856746688500020 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 401435664 T + 127868030128292686 T^{2} - 401435664 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 30314907095525540 T^{2} + \)\(69\!\cdots\!38\)\( T^{4} - 30314907095525540 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6539040 p T + 303933580876193438 T^{2} + 6539040 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 597414961848210420 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - 597414961848210420 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 138178380 T + 45471108987586518 T^{2} + 138178380 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 228500609440802180 T^{2} - \)\(20\!\cdots\!22\)\( T^{4} - 228500609440802180 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80038183841276154018164380699, −13.38092702072460507695123610426, −12.82335275556508879722855856083, −12.68211263888722155807232384299, −12.08183471228442366254244728310, −11.74836181911724604563497968015, −11.49999631144230547647770843971, −10.59987351316508526595084221549, −10.53502779041389204895393382805, −9.993722822478043476379211024327, −9.582877331672943819847063024252, −8.788909222408938161821611378684, −8.237634649708821952232549981708, −7.952071566358300412624674409202, −7.952005287620634517346020789149, −7.06835646614364493304219986721, −6.50605299324586431243291332122, −5.53510202738511668884936437692, −5.03399840262277744389502857975, −4.33547937838848091603762914151, −4.19085699587622456471185481557, −2.93958597831388971195139752913, −2.76927623526238644301959232882, −0.828568064698403981636625765331, −0.52097520935218875100403050296,
0.52097520935218875100403050296, 0.828568064698403981636625765331, 2.76927623526238644301959232882, 2.93958597831388971195139752913, 4.19085699587622456471185481557, 4.33547937838848091603762914151, 5.03399840262277744389502857975, 5.53510202738511668884936437692, 6.50605299324586431243291332122, 7.06835646614364493304219986721, 7.952005287620634517346020789149, 7.952071566358300412624674409202, 8.237634649708821952232549981708, 8.788909222408938161821611378684, 9.582877331672943819847063024252, 9.993722822478043476379211024327, 10.53502779041389204895393382805, 10.59987351316508526595084221549, 11.49999631144230547647770843971, 11.74836181911724604563497968015, 12.08183471228442366254244728310, 12.68211263888722155807232384299, 12.82335275556508879722855856083, 13.38092702072460507695123610426, 13.80038183841276154018164380699
