| L(s) = 1 | + 86·3-s + 128·4-s + 2.18e3·9-s + 2.64e4·11-s + 1.10e4·12-s + 1.19e5·19-s + 1.56e5·25-s − 7.18e4·27-s + 2.27e6·33-s + 2.79e5·36-s + 7.10e5·41-s − 2.20e5·43-s + 3.38e6·44-s − 1.64e6·49-s + 1.02e7·57-s − 3.09e6·59-s − 2.09e6·64-s − 3.85e6·67-s + 9.73e6·73-s + 1.34e7·75-s + 1.52e7·76-s − 6.17e6·81-s − 9.93e6·97-s + 5.78e7·99-s + 2.00e7·100-s − 9.19e6·108-s + 3.68e8·121-s + ⋯ |
| L(s) = 1 | + 1.83·3-s + 4-s + 9-s + 5.98·11-s + 1.83·12-s + 3.99·19-s + 2·25-s − 0.702·27-s + 11.0·33-s + 36-s + 1.61·41-s − 0.422·43-s + 5.98·44-s − 2·49-s + 7.34·57-s − 1.96·59-s − 64-s − 1.56·67-s + 2.92·73-s + 3.67·75-s + 3.99·76-s − 1.29·81-s − 1.10·97-s + 5.98·99-s + 2·100-s − 0.702·108-s + 18.9·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(39.89126823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(39.89126823\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 86 T + 5209 T^{2} - 86 p^{7} T^{3} + p^{14} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8814 T + p^{7} T^{2} )^{2}( 1 - 8814 T + 58199425 T^{2} - 8814 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22182 T + 81702451 T^{2} - 22182 p^{7} T^{3} + p^{14} T^{4} )( 1 + 22182 T + 81702451 T^{2} + 22182 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 59722 T + 2672845545 T^{2} - 59722 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 236886 T + p^{7} T^{2} )^{2}( 1 - 236886 T - 138639296885 T^{2} - 236886 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 220510 T + p^{7} T^{2} )^{2}( 1 - 220510 T - 223193951007 T^{2} - 220510 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1030926 T + p^{7} T^{2} )^{2}( 1 + 1030926 T - 1425843067343 T^{2} + 1030926 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3851302 T + p^{7} T^{2} )^{2}( 1 - 3851302 T + 8771815489881 T^{2} - 3851302 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 4865614 T + 12626801077899 T^{2} - 4865614 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4808934 T - 4010204773271 T^{2} - 4808934 p^{7} T^{3} + p^{14} T^{4} )( 1 + 4808934 T - 4010204773271 T^{2} + 4808934 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 7073118 T + p^{7} T^{2} )^{2}( 1 + 7073118 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9938890 T + p^{7} T^{2} )^{2}( 1 - 9938890 T + 17983249953987 T^{2} - 9938890 p^{7} T^{3} + p^{14} T^{4} ) \) |
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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
−9.436103661002744624143126967629, −9.212794169112183583294814465348, −8.787735341831990155217067367284, −8.452470490247965684155252256184, −8.401569913921894218934222222443, −7.68467485963692991270520101243, −7.29596410674647825813593361808, −7.19099056274513325517734082660, −7.00042817194734607871508430700, −6.34652734264972349476980400951, −6.28827585123594796128231943562, −6.22686504063554157609741425721, −5.45324432761821132225210075626, −4.91960026086787666065580647978, −4.48438450234708353311861552860, −4.01866334637567175363932493556, −3.77493392911711761742846270637, −3.20915436999931203413967186192, −3.09116906566398791224464295541, −3.08321737735635882748448955492, −2.07176341725895568888870341981, −1.63483072558830432818538792385, −1.37290450211577231122732501750, −0.988888052446850999153925741886, −0.875158607874203256893342963208,
0.875158607874203256893342963208, 0.988888052446850999153925741886, 1.37290450211577231122732501750, 1.63483072558830432818538792385, 2.07176341725895568888870341981, 3.08321737735635882748448955492, 3.09116906566398791224464295541, 3.20915436999931203413967186192, 3.77493392911711761742846270637, 4.01866334637567175363932493556, 4.48438450234708353311861552860, 4.91960026086787666065580647978, 5.45324432761821132225210075626, 6.22686504063554157609741425721, 6.28827585123594796128231943562, 6.34652734264972349476980400951, 7.00042817194734607871508430700, 7.19099056274513325517734082660, 7.29596410674647825813593361808, 7.68467485963692991270520101243, 8.401569913921894218934222222443, 8.452470490247965684155252256184, 8.787735341831990155217067367284, 9.212794169112183583294814465348, 9.436103661002744624143126967629
