| L(s) = 1 | + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯ |
| L(s) = 1 | + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01410600819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01410600819\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 3 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 5 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 13 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 31 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 37 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 43 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 47 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 53 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 59 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 61 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 71 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
| 73 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 89 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 97 | \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.40598913735417001545858206503, −2.23662004123553542176174588292, −2.23650065638581581160866804457, −2.07114346706589476641891385131, −2.06548028031878964011245206911, −2.06059751667065391828083876081, −1.98029049798854586750929716549, −1.86561170943081062170291618874, −1.84546903389970258791212010733, −1.83874668153421823065654875332, −1.75390355149563486657709909846, −1.64217527698553263815979520721, −1.58511757954490734084886290531, −1.44555140690865457603447390606, −1.40013686751848431108800247719, −1.39127232227049611450809539498, −1.33702279595459545319028097007, −1.31359029467698936436623054699, −0.981635519143627430834342897639, −0.951139192621951683066811718955, −0.872889831212535254413303641556, −0.812435283290659250667624295587, −0.67298068428479325757427945693, −0.54819995392615220150138307518, −0.094203593162691583514429885827,
0.094203593162691583514429885827, 0.54819995392615220150138307518, 0.67298068428479325757427945693, 0.812435283290659250667624295587, 0.872889831212535254413303641556, 0.951139192621951683066811718955, 0.981635519143627430834342897639, 1.31359029467698936436623054699, 1.33702279595459545319028097007, 1.39127232227049611450809539498, 1.40013686751848431108800247719, 1.44555140690865457603447390606, 1.58511757954490734084886290531, 1.64217527698553263815979520721, 1.75390355149563486657709909846, 1.83874668153421823065654875332, 1.84546903389970258791212010733, 1.86561170943081062170291618874, 1.98029049798854586750929716549, 2.06059751667065391828083876081, 2.06548028031878964011245206911, 2.07114346706589476641891385131, 2.23650065638581581160866804457, 2.23662004123553542176174588292, 2.40598913735417001545858206503
Plot not available for L-functions of degree greater than 10.
