| L(s) = 1 | + 748·4-s + 1.00e5·7-s − 2.07e6·13-s + 4.19e6·16-s − 2.31e6·19-s + 9.73e7·25-s + 7.49e7·28-s + 1.78e8·31-s − 2.26e9·37-s − 2.51e9·43-s + 6.46e9·49-s − 1.55e9·52-s + 2.22e9·61-s + 8.99e9·64-s − 8.06e9·67-s − 1.24e11·73-s − 1.73e9·76-s + 1.23e9·79-s − 2.08e11·91-s + 2.09e11·97-s + 7.27e10·100-s + 1.24e10·103-s + 9.67e10·109-s + 4.20e11·112-s + 2.65e11·121-s + 1.33e11·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.365·4-s + 2.25·7-s − 1.55·13-s + 16-s − 0.214·19-s + 1.99·25-s + 0.822·28-s + 1.11·31-s − 5.37·37-s − 2.60·43-s + 3.26·49-s − 0.566·52-s + 0.337·61-s + 1.04·64-s − 0.730·67-s − 7.03·73-s − 0.0784·76-s + 0.0451·79-s − 3.49·91-s + 2.47·97-s + 0.727·100-s + 0.106·103-s + 0.602·109-s + 2.25·112-s + 0.928·121-s + 0.408·124-s − 0.483·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.9857067559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9857067559\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 187 p^{2} T^{2} - 227175 p^{4} T^{4} - 187 p^{24} T^{6} + p^{44} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 3892858 p^{2} T^{2} + 11339646142539 p^{4} T^{4} - 3892858 p^{24} T^{6} + p^{44} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 7153 p T + 10811802 p^{2} T^{2} - 7153 p^{12} T^{3} + p^{22} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 265021530454 T^{2} - \)\(11\!\cdots\!05\)\( T^{4} - 265021530454 p^{22} T^{6} + p^{44} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 1039121 T - 712387941396 T^{2} + 1039121 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 34457089469074 T^{2} + p^{22} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 579301 T + p^{11} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 1858471075150654 T^{2} + \)\(25\!\cdots\!87\)\( T^{4} - 1858471075150654 p^{22} T^{6} + p^{44} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 9824585799078710 T^{2} - \)\(52\!\cdots\!41\)\( T^{4} + 9824585799078710 p^{22} T^{6} + p^{44} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 89057548 T - 17477230040632527 T^{2} - 89057548 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 566931913 T + p^{11} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 706607837206255886 T^{2} + \)\(19\!\cdots\!15\)\( T^{4} + 706607837206255886 p^{22} T^{6} + p^{44} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 1257516392 T + 652053736677474957 T^{2} + 1257516392 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3511003396686462386 T^{2} + \)\(62\!\cdots\!87\)\( T^{4} + 3511003396686462386 p^{22} T^{6} + p^{44} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 5339277554304352358 T^{2} + p^{22} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 48154397067700509046 T^{2} + \)\(14\!\cdots\!35\)\( T^{4} - 48154397067700509046 p^{22} T^{6} + p^{44} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 1112205613 T - 42276916285847132892 T^{2} - 1112205613 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 4034467871 T - \)\(10\!\cdots\!42\)\( T^{2} + 4034467871 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + \)\(41\!\cdots\!30\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 31135508287 T + p^{11} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 617559745 T - \)\(74\!\cdots\!54\)\( T^{2} - 617559745 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - \)\(25\!\cdots\!86\)\( T^{2} + \)\(48\!\cdots\!07\)\( T^{4} - \)\(25\!\cdots\!86\)\( p^{22} T^{6} + p^{44} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + \)\(55\!\cdots\!78\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 104737106371 T + \)\(38\!\cdots\!88\)\( T^{2} - 104737106371 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
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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
−8.126034122120182089795973542189, −8.096086203199243508249361290065, −7.902025992954092395154787386017, −7.17891767701616184492510153013, −7.11814617117900682299661351782, −7.02204276763378495142909662135, −6.70756871895938416717353019901, −6.17488781727437302767974928422, −5.55287932993197957004801885581, −5.50281206793209193643251865732, −5.17087090445751150847776351252, −4.85106288970882569206618175663, −4.72612633211159432942946051839, −4.30188213819053749638247137480, −4.01907666927115817959310725202, −3.13922510118764307589816772053, −3.12700488661052984136522749803, −3.06379053408607573104105014168, −2.28230873415437732183259044939, −1.81094890712282814869582017808, −1.77796417336346660878268750226, −1.48833950679086131310699478194, −1.11250365135016108102095108102, −0.60664228057633326924061872340, −0.10731586098279641988095603605,
0.10731586098279641988095603605, 0.60664228057633326924061872340, 1.11250365135016108102095108102, 1.48833950679086131310699478194, 1.77796417336346660878268750226, 1.81094890712282814869582017808, 2.28230873415437732183259044939, 3.06379053408607573104105014168, 3.12700488661052984136522749803, 3.13922510118764307589816772053, 4.01907666927115817959310725202, 4.30188213819053749638247137480, 4.72612633211159432942946051839, 4.85106288970882569206618175663, 5.17087090445751150847776351252, 5.50281206793209193643251865732, 5.55287932993197957004801885581, 6.17488781727437302767974928422, 6.70756871895938416717353019901, 7.02204276763378495142909662135, 7.11814617117900682299661351782, 7.17891767701616184492510153013, 7.902025992954092395154787386017, 8.096086203199243508249361290065, 8.126034122120182089795973542189
