| L(s) = 1 | + 1.03e4·7-s + 4.10e5·13-s + 2.62e6·19-s + 3.51e6·25-s + 2.36e6·31-s − 7.98e7·37-s − 3.75e8·43-s − 1.00e9·49-s − 3.27e9·61-s − 2.39e9·67-s − 1.16e10·73-s − 5.08e9·79-s + 4.23e9·91-s + 1.07e10·97-s + 4.00e9·103-s + 1.52e10·109-s + 4.08e10·121-s + 127-s + 131-s + 2.70e10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.614·7-s + 1.10·13-s + 1.06·19-s + 0.360·25-s + 0.0826·31-s − 1.15·37-s − 2.55·43-s − 3.56·49-s − 3.87·61-s − 1.77·67-s − 5.63·73-s − 1.65·79-s + 0.678·91-s + 1.25·97-s + 0.345·103-s + 0.990·109-s + 1.57·121-s + 0.651·133-s − 2.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.06803376611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06803376611\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 703392 p T^{2} + 15307357106 p^{2} T^{4} - 703392 p^{21} T^{6} + p^{40} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 5160 T + 77538494 p T^{2} - 5160 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 40884071460 T^{2} + \)\(11\!\cdots\!62\)\( T^{4} - 40884071460 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 205184 T + 237766365522 T^{2} - 205184 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3932813310336 T^{2} + \)\(92\!\cdots\!26\)\( T^{4} - 3932813310336 p^{20} T^{6} + p^{40} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 1312864 T + 10166414343186 T^{2} - 1312864 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 7725264569340 T^{2} - \)\(49\!\cdots\!38\)\( T^{4} + 7725264569340 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1629929609373088 T^{2} + \)\(10\!\cdots\!98\)\( T^{4} - 1629929609373088 p^{20} T^{6} + p^{40} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 1182664 T + 1365153842591826 T^{2} - 1182664 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 39942468 T + 9291618286927094 T^{2} + 39942468 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 43447070102894080 T^{2} + \)\(81\!\cdots\!42\)\( T^{4} - 43447070102894080 p^{20} T^{6} + p^{40} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 187833328 T + 52029733619543154 T^{2} + 187833328 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 154991949479723332 T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - 154991949479723332 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 332888678286636192 T^{2} + \)\(88\!\cdots\!58\)\( T^{4} - 332888678286636192 p^{20} T^{6} + p^{40} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1285501985600873700 T^{2} + \)\(91\!\cdots\!42\)\( T^{4} - 1285501985600873700 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 1636782364 T + 1904076282513860886 T^{2} + 1636782364 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 1195499408 T + 2301363399536144274 T^{2} + 1195499408 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 461429486652630660 T^{2} + \)\(18\!\cdots\!62\)\( T^{4} - 461429486652630660 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 5843600224 T + 16237152225356142882 T^{2} + 5843600224 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2540132488 T + 20200211425431361938 T^{2} + 2540132488 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12596088612044915940 T^{2} + \)\(60\!\cdots\!78\)\( p^{2} T^{4} - 12596088612044915940 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 91237187160550084864 T^{2} + \)\(37\!\cdots\!26\)\( T^{4} - 91237187160550084864 p^{20} T^{6} + p^{40} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 5381750080 T + 14367429702575364738 T^{2} - 5381750080 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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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
−8.668386784045428463213615286783, −8.532453216953538795881945946503, −8.230155393409260744626584290486, −7.76250299426302603478189774808, −7.30589001233006870208368546886, −7.22460055727949050666558996146, −7.06328170324646450392933316240, −6.29683799286578252372753712239, −6.18393802573162623927094018165, −5.78772726939281019736257442560, −5.76164558800883489328783285590, −4.96282728160922698881818434899, −4.74369599607600269632858248415, −4.57677175239904072652012323252, −4.34634657371771058950491650079, −3.48509686739591939513890623523, −3.26928383491850337636369759856, −3.05678751059599772648932962314, −2.97108711963808293714990262164, −1.90383420159163850449272962749, −1.71590260865811683120636661155, −1.39075300412921517310018716283, −1.37787122066436818825744101633, −0.56822877424044987980809507192, −0.03574404450340139704595641383,
0.03574404450340139704595641383, 0.56822877424044987980809507192, 1.37787122066436818825744101633, 1.39075300412921517310018716283, 1.71590260865811683120636661155, 1.90383420159163850449272962749, 2.97108711963808293714990262164, 3.05678751059599772648932962314, 3.26928383491850337636369759856, 3.48509686739591939513890623523, 4.34634657371771058950491650079, 4.57677175239904072652012323252, 4.74369599607600269632858248415, 4.96282728160922698881818434899, 5.76164558800883489328783285590, 5.78772726939281019736257442560, 6.18393802573162623927094018165, 6.29683799286578252372753712239, 7.06328170324646450392933316240, 7.22460055727949050666558996146, 7.30589001233006870208368546886, 7.76250299426302603478189774808, 8.230155393409260744626584290486, 8.532453216953538795881945946503, 8.668386784045428463213615286783
