L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 3·9-s + 2·11-s + 2·12-s − 12·17-s + 3·18-s − 19-s + 2·22-s − 2·25-s + 2·27-s + 4·33-s − 12·34-s + 3·36-s − 38-s + 2·41-s + 2·43-s + 2·44-s + 49-s − 2·50-s − 24·51-s + 2·54-s − 2·57-s − 9·59-s + 4·66-s + ⋯ |
L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 3·9-s + 2·11-s + 2·12-s − 12·17-s + 3·18-s − 19-s + 2·22-s − 2·25-s + 2·27-s + 4·33-s − 12·34-s + 3·36-s − 38-s + 2·41-s + 2·43-s + 2·44-s + 49-s − 2·50-s − 24·51-s + 2·54-s − 2·57-s − 9·59-s + 4·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1002752062\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1002752062\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
good | 3 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 7 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 11 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 13 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 17 | \( ( 1 + T + T^{2} )^{10}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 19 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 23 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 29 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 43 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 59 | \( ( 1 + T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 71 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 79 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 83 | \( ( 1 + T + T^{2} )^{10}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
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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
−2.80311667849474606731546959897, −2.76456282467137060877622230801, −2.66592575198728962847384038119, −2.65059047860534385449777820188, −2.59319488173354515801319615532, −2.47084844526413006591070800537, −2.44789873355177285875543061234, −2.41811566403519489068148741322, −2.35190038931418778041758521939, −2.18691422719419511696280571902, −2.15532631033475683501512479779, −2.15033179344780173187865113960, −2.10281290068202996215980681425, −2.06538464953225577092620536470, −1.67418817551595387719434100082, −1.64022771488741472356352291560, −1.60152759867534834100729695222, −1.58151440606263223617628536540, −1.54038099917372231128586946478, −1.53947678530118681467843482374, −1.44849931297916636453214105704, −1.41551709662210133830461713591, −1.40486506447869574628400611934, −1.05623195569483194404743320644, −0.47191492857109870696145206375,
0.47191492857109870696145206375, 1.05623195569483194404743320644, 1.40486506447869574628400611934, 1.41551709662210133830461713591, 1.44849931297916636453214105704, 1.53947678530118681467843482374, 1.54038099917372231128586946478, 1.58151440606263223617628536540, 1.60152759867534834100729695222, 1.64022771488741472356352291560, 1.67418817551595387719434100082, 2.06538464953225577092620536470, 2.10281290068202996215980681425, 2.15033179344780173187865113960, 2.15532631033475683501512479779, 2.18691422719419511696280571902, 2.35190038931418778041758521939, 2.41811566403519489068148741322, 2.44789873355177285875543061234, 2.47084844526413006591070800537, 2.59319488173354515801319615532, 2.65059047860534385449777820188, 2.66592575198728962847384038119, 2.76456282467137060877622230801, 2.80311667849474606731546959897
Plot not available for L-functions of degree greater than 10.