| L(s) = 1 | − 3-s + 2·7-s + 2·9-s − 2·19-s − 2·21-s + 18·25-s − 5·27-s + 6·29-s + 4·41-s − 20·43-s − 25·49-s + 26·53-s + 2·57-s + 2·59-s − 4·61-s + 4·63-s + 8·71-s − 26·73-s − 18·75-s + 11·81-s − 6·87-s − 4·89-s + 54·107-s + 60·113-s + 60·121-s − 4·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 2/3·9-s − 0.458·19-s − 0.436·21-s + 18/5·25-s − 0.962·27-s + 1.11·29-s + 0.624·41-s − 3.04·43-s − 3.57·49-s + 3.57·53-s + 0.264·57-s + 0.260·59-s − 0.512·61-s + 0.503·63-s + 0.949·71-s − 3.04·73-s − 2.07·75-s + 11/9·81-s − 0.643·87-s − 0.423·89-s + 5.22·107-s + 5.64·113-s + 5.45·121-s − 0.360·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.300288757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.300288757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T - T^{2} + 2 T^{3} - 2 T^{4} - 22 T^{5} - 2 p T^{6} + 2 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | \( 1 + 2 T + 3 T^{2} - 80 T^{3} + 10 p T^{4} + 20 p T^{5} + 10 p^{2} T^{6} - 80 p^{2} T^{7} + 3 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 5 | \( 1 - 18 T^{2} + 146 T^{4} - 904 T^{6} + 1241 p T^{8} - 36972 T^{10} + 1241 p^{3} T^{12} - 904 p^{4} T^{14} + 146 p^{6} T^{16} - 18 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( ( 1 - T + 2 p T^{2} - 27 T^{3} + 149 T^{4} - 216 T^{5} + 149 p T^{6} - 27 p^{2} T^{7} + 2 p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( 1 - 60 T^{2} + 1934 T^{4} - 3790 p T^{6} + 668209 T^{8} - 8283636 T^{10} + 668209 p^{2} T^{12} - 3790 p^{5} T^{14} + 1934 p^{6} T^{16} - 60 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( 1 - 73 T^{2} + 207 p T^{4} - 67108 T^{6} + 1259228 T^{8} - 18454230 T^{10} + 1259228 p^{2} T^{12} - 67108 p^{4} T^{14} + 207 p^{7} T^{16} - 73 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 69 T^{2} + 2660 T^{4} - 70419 T^{6} + 1483831 T^{8} - 26674760 T^{10} + 1483831 p^{2} T^{12} - 70419 p^{4} T^{14} + 2660 p^{6} T^{16} - 69 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( 1 - 101 T^{2} + 5383 T^{4} - 206040 T^{6} + 6263624 T^{8} - 157424902 T^{10} + 6263624 p^{2} T^{12} - 206040 p^{4} T^{14} + 5383 p^{6} T^{16} - 101 p^{8} T^{18} + p^{10} T^{20} \) |
| 29 | \( ( 1 - 3 T + 93 T^{2} - 208 T^{3} + 4494 T^{4} - 8234 T^{5} + 4494 p T^{6} - 208 p^{2} T^{7} + 93 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( 1 - 176 T^{2} + 16041 T^{4} - 990384 T^{6} + 45304566 T^{8} - 1593191168 T^{10} + 45304566 p^{2} T^{12} - 990384 p^{4} T^{14} + 16041 p^{6} T^{16} - 176 p^{8} T^{18} + p^{10} T^{20} \) |
| 37 | \( 1 - 196 T^{2} + 20993 T^{4} - 1512784 T^{6} + 81431766 T^{8} - 3393790104 T^{10} + 81431766 p^{2} T^{12} - 1512784 p^{4} T^{14} + 20993 p^{6} T^{16} - 196 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 - 2 T + 147 T^{2} - 380 T^{3} + 9932 T^{4} - 24180 T^{5} + 9932 p T^{6} - 380 p^{2} T^{7} + 147 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( ( 1 + 10 T + 156 T^{2} + 1104 T^{3} + 9603 T^{4} + 57532 T^{5} + 9603 p T^{6} + 1104 p^{2} T^{7} + 156 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 47 | \( 1 - 266 T^{2} + 34794 T^{4} - 3034216 T^{6} + 198749709 T^{8} - 10364810252 T^{10} + 198749709 p^{2} T^{12} - 3034216 p^{4} T^{14} + 34794 p^{6} T^{16} - 266 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( ( 1 - 13 T + 243 T^{2} - 1958 T^{3} + 22352 T^{4} - 135946 T^{5} + 22352 p T^{6} - 1958 p^{2} T^{7} + 243 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 59 | \( ( 1 - T + 89 T^{2} - 194 T^{3} + 6044 T^{4} + 2406 T^{5} + 6044 p T^{6} - 194 p^{2} T^{7} + 89 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 61 | \( ( 1 + 2 T + 182 T^{2} + 504 T^{3} + 17033 T^{4} + 50092 T^{5} + 17033 p T^{6} + 504 p^{2} T^{7} + 182 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 199 T^{2} + 24849 T^{4} - 2435340 T^{6} + 208550550 T^{8} - 14951642938 T^{10} + 208550550 p^{2} T^{12} - 2435340 p^{4} T^{14} + 24849 p^{6} T^{16} - 199 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 - 4 T + 161 T^{2} - 1184 T^{3} + 13600 T^{4} - 135736 T^{5} + 13600 p T^{6} - 1184 p^{2} T^{7} + 161 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( ( 1 + 13 T + 290 T^{2} + 1959 T^{3} + 27779 T^{4} + 133708 T^{5} + 27779 p T^{6} + 1959 p^{2} T^{7} + 290 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 79 | \( 1 - 304 T^{2} + 34229 T^{4} - 1188496 T^{6} - 113976918 T^{8} + 16630061952 T^{10} - 113976918 p^{2} T^{12} - 1188496 p^{4} T^{14} + 34229 p^{6} T^{16} - 304 p^{8} T^{18} + p^{10} T^{20} \) |
| 83 | \( 1 - 590 T^{2} + 168197 T^{4} - 30643848 T^{6} + 3958839394 T^{8} - 379176726036 T^{10} + 3958839394 p^{2} T^{12} - 30643848 p^{4} T^{14} + 168197 p^{6} T^{16} - 590 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 + 2 T + 223 T^{2} - 332 T^{3} + 22920 T^{4} - 90348 T^{5} + 22920 p T^{6} - 332 p^{2} T^{7} + 223 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 350 T^{2} + 53073 T^{4} - 4388768 T^{6} + 168448558 T^{8} - 2002086788 T^{10} + 168448558 p^{2} T^{12} - 4388768 p^{4} T^{14} + 53073 p^{6} T^{16} - 350 p^{8} T^{18} + p^{10} T^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.61023890827521913035709477645, −3.60578163017445106879068104035, −3.48523144880268599955669948334, −3.24268220566597406072366448120, −3.19914615817306007526637175579, −3.17444952280101372096551075926, −3.09943211819160921456116907504, −2.88554889146180247884406554030, −2.88163392737264686810146731832, −2.67439336218025617277021392097, −2.53456709147598294679952772385, −2.47532752658513390089856971530, −2.12285265911615744198150067071, −2.06767602408533053114108521055, −1.89367377645874448213238401996, −1.76374847081236126400185535220, −1.75642424146724459559983209527, −1.68152417165251832994400682939, −1.63241319103631849265549376681, −1.01482224455774604841194560288, −0.951996735972667854781308818851, −0.947848502445724987094186728531, −0.76010250825886523250787418444, −0.60286678488921015899669574105, −0.19022532742368211677500864059,
0.19022532742368211677500864059, 0.60286678488921015899669574105, 0.76010250825886523250787418444, 0.947848502445724987094186728531, 0.951996735972667854781308818851, 1.01482224455774604841194560288, 1.63241319103631849265549376681, 1.68152417165251832994400682939, 1.75642424146724459559983209527, 1.76374847081236126400185535220, 1.89367377645874448213238401996, 2.06767602408533053114108521055, 2.12285265911615744198150067071, 2.47532752658513390089856971530, 2.53456709147598294679952772385, 2.67439336218025617277021392097, 2.88163392737264686810146731832, 2.88554889146180247884406554030, 3.09943211819160921456116907504, 3.17444952280101372096551075926, 3.19914615817306007526637175579, 3.24268220566597406072366448120, 3.48523144880268599955669948334, 3.60578163017445106879068104035, 3.61023890827521913035709477645
Plot not available for L-functions of degree greater than 10.
