L(s) = 1 | − 2·2-s + 3·4-s − 2·8-s − 9-s + 16-s + 2·18-s + 23-s + 25-s − 4·29-s − 3·36-s − 2·46-s − 2·50-s + 8·58-s − 4·71-s + 2·72-s + 81-s + 3·92-s + 3·100-s − 11·109-s − 12·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 8·142-s − 144-s + ⋯ |
L(s) = 1 | − 2·2-s + 3·4-s − 2·8-s − 9-s + 16-s + 2·18-s + 23-s + 25-s − 4·29-s − 3·36-s − 2·46-s − 2·50-s + 8·58-s − 4·71-s + 2·72-s + 81-s + 3·92-s + 3·100-s − 11·109-s − 12·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 8·142-s − 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{40} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{40} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05604940699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05604940699\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
good | 2 | \( ( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} )^{2} \) |
| 3 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 11 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 13 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 31 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{10} \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 59 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 67 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 71 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \) |
| 73 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} - T^{22} + T^{26} + T^{28} - T^{32} - T^{34} + T^{38} + T^{40} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} - T^{11} + T^{13} + T^{14} - T^{16} - T^{17} + T^{19} + T^{20} ) \) |
| 97 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
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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
−2.46848775869645938138212753208, −2.43790303669818373192264807247, −2.41880347257028062385771850318, −2.24708366591360727879392779405, −2.13005598575029814602271546203, −2.08393364318807151459821459946, −1.99346829424782372295086047028, −1.87771475905449857482971404226, −1.84554381322945571023078866229, −1.83933244568965573789923050009, −1.67383471708748235781728371884, −1.64157538087738616863094823087, −1.63682824379856793164577380908, −1.62476182503395644162606490316, −1.49228168730752668457043169596, −1.47514933802020632595508609688, −1.37379088010298882728531123747, −1.18288789182723832720710222437, −1.14863134526262189418523393129, −1.12634725602528441627083184747, −0.953890809158487010274636955666, −0.937083859639225685074738058789, −0.900762185689583992017170932007, −0.68759696971867601346627825010, −0.19768603830847262584629379663,
0.19768603830847262584629379663, 0.68759696971867601346627825010, 0.900762185689583992017170932007, 0.937083859639225685074738058789, 0.953890809158487010274636955666, 1.12634725602528441627083184747, 1.14863134526262189418523393129, 1.18288789182723832720710222437, 1.37379088010298882728531123747, 1.47514933802020632595508609688, 1.49228168730752668457043169596, 1.62476182503395644162606490316, 1.63682824379856793164577380908, 1.64157538087738616863094823087, 1.67383471708748235781728371884, 1.83933244568965573789923050009, 1.84554381322945571023078866229, 1.87771475905449857482971404226, 1.99346829424782372295086047028, 2.08393364318807151459821459946, 2.13005598575029814602271546203, 2.24708366591360727879392779405, 2.41880347257028062385771850318, 2.43790303669818373192264807247, 2.46848775869645938138212753208
Plot not available for L-functions of degree greater than 10.