| L(s) = 1 | + 8·5-s + 40·13-s + 24·17-s + 36·25-s + 40·29-s + 72·37-s + 88·41-s + 68·49-s − 24·53-s − 56·61-s + 320·65-s − 120·73-s + 192·85-s + 312·89-s − 248·97-s − 664·101-s − 24·109-s − 328·113-s + 260·121-s + 312·125-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 3.07·13-s + 1.41·17-s + 1.43·25-s + 1.37·29-s + 1.94·37-s + 2.14·41-s + 1.38·49-s − 0.452·53-s − 0.918·61-s + 4.92·65-s − 1.64·73-s + 2.25·85-s + 3.50·89-s − 2.55·97-s − 6.57·101-s − 0.220·109-s − 2.90·113-s + 2.14·121-s + 2.49·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.329210748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.329210748\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 4 T + 6 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2886 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 33894 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 422 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 159654 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 994 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 36 T + 62 p T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 44 T + 3654 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1796 T^{2} - 35994 T^{4} - 1796 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2180 T^{2} + 8539014 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 5606 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 462054 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 11204 T^{2} + 62050854 T^{4} - 11204 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 60 T + 8486 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21188 T^{2} + 206547366 T^{4} - 21188 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 124 T + 15750 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.474829720227529053596098133405, −8.015302184175872314309093117284, −7.86743083734058447621667776122, −7.83935652901921888690548256736, −7.33589218495314541913050033139, −6.96409727515459730060680387228, −6.58265831598696639274559606554, −6.43857978292357709983024846819, −6.31227445085428122384410689138, −5.89230379643022155001426482484, −5.70948458436261223651329980887, −5.51310601354815687225015346981, −5.45591254331241313178869422380, −4.83752650241779554639991420929, −4.30580536817979001386069394196, −4.24220151727672429232483847798, −3.98234674621831428417294503909, −3.49279854997097632085256647252, −3.06960892795644005561749396314, −2.71957424196197259997271140055, −2.63199227613954430545955762714, −1.87698726339158728145293826524, −1.36989869128918194192688543692, −1.15078906162402368572529856725, −0.812795444008315609737084899139,
0.812795444008315609737084899139, 1.15078906162402368572529856725, 1.36989869128918194192688543692, 1.87698726339158728145293826524, 2.63199227613954430545955762714, 2.71957424196197259997271140055, 3.06960892795644005561749396314, 3.49279854997097632085256647252, 3.98234674621831428417294503909, 4.24220151727672429232483847798, 4.30580536817979001386069394196, 4.83752650241779554639991420929, 5.45591254331241313178869422380, 5.51310601354815687225015346981, 5.70948458436261223651329980887, 5.89230379643022155001426482484, 6.31227445085428122384410689138, 6.43857978292357709983024846819, 6.58265831598696639274559606554, 6.96409727515459730060680387228, 7.33589218495314541913050033139, 7.83935652901921888690548256736, 7.86743083734058447621667776122, 8.015302184175872314309093117284, 8.474829720227529053596098133405
