| L(s) = 1 | − 2·4-s − 5·16-s + 24·17-s + 8·25-s − 12·29-s + 24·31-s − 12·37-s + 12·41-s + 12·49-s + 20·64-s − 24·67-s − 48·68-s − 24·83-s + 4·97-s − 16·100-s + 24·101-s + 16·103-s + 48·107-s + 24·116-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 24·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 5/4·16-s + 5.82·17-s + 8/5·25-s − 2.22·29-s + 4.31·31-s − 1.97·37-s + 1.87·41-s + 12/7·49-s + 5/2·64-s − 2.93·67-s − 5.82·68-s − 2.63·83-s + 0.406·97-s − 8/5·100-s + 2.38·101-s + 1.57·103-s + 4.64·107-s + 2.22·116-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.129641215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.129641215\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 191 T^{4} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1110 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 4166 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15591 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 15366 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 120 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23574 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 24866 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19814 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 24519 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
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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
−7.15901030155878407331719741345, −6.81827449628234482064041230439, −6.78852667116849968258384100683, −6.15147375137983256210614214213, −6.13231009072525667738771282386, −5.87518275545777147718248524483, −5.66003355945651294873628852463, −5.57925922923502941948474451950, −5.27714493000366168947779133600, −4.96355565654586133239733407044, −4.68136264972373528801725023166, −4.61821961762401860156742153419, −4.44736280508665472381241058720, −3.87557070806318009246896135588, −3.83805922632564112381975736196, −3.49383957047404773436491444878, −3.13566021778699767677843421811, −3.01199844647480961693460681155, −2.87020849028568564162484460752, −2.50202572027507866532171620786, −1.77042917205826519928183409879, −1.75768412606207051868366068912, −1.05063964477033145930900136371, −0.816250183710423489464148460934, −0.68230968651203488465923674079,
0.68230968651203488465923674079, 0.816250183710423489464148460934, 1.05063964477033145930900136371, 1.75768412606207051868366068912, 1.77042917205826519928183409879, 2.50202572027507866532171620786, 2.87020849028568564162484460752, 3.01199844647480961693460681155, 3.13566021778699767677843421811, 3.49383957047404773436491444878, 3.83805922632564112381975736196, 3.87557070806318009246896135588, 4.44736280508665472381241058720, 4.61821961762401860156742153419, 4.68136264972373528801725023166, 4.96355565654586133239733407044, 5.27714493000366168947779133600, 5.57925922923502941948474451950, 5.66003355945651294873628852463, 5.87518275545777147718248524483, 6.13231009072525667738771282386, 6.15147375137983256210614214213, 6.78852667116849968258384100683, 6.81827449628234482064041230439, 7.15901030155878407331719741345
