| L(s) = 1 | − 2·3-s + 6·4-s + 3·5-s + 6·7-s + 9-s + 2·11-s − 12·12-s − 6·13-s − 6·15-s + 12·16-s + 2·17-s − 19-s + 18·20-s − 12·21-s + 3·25-s + 2·27-s + 36·28-s − 7·29-s − 2·31-s − 4·33-s + 18·35-s + 6·36-s + 8·37-s + 12·39-s − 10·41-s − 24·43-s + 12·44-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3·4-s + 1.34·5-s + 2.26·7-s + 1/3·9-s + 0.603·11-s − 3.46·12-s − 1.66·13-s − 1.54·15-s + 3·16-s + 0.485·17-s − 0.229·19-s + 4.02·20-s − 2.61·21-s + 3/5·25-s + 0.384·27-s + 6.80·28-s − 1.29·29-s − 0.359·31-s − 0.696·33-s + 3.04·35-s + 36-s + 1.31·37-s + 1.92·39-s − 1.56·41-s − 3.65·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.798874583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.798874583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 - T + T^{2} )^{3} \) |
| 7 | \( ( 1 - T )^{6} \) |
| 19 | \( 1 + T + 20 T^{2} + 109 T^{3} + 20 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \) |
| 3 | \( ( 1 + T - 2 T^{2} - p T^{3} - 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )( 1 + T + 4 T^{2} + p T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} ) \) |
| 11 | \( ( 1 - T + 12 T^{2} - 25 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \) |
| 17 | \( 1 - 2 T - 11 T^{2} + 154 T^{3} - 214 T^{4} - 938 T^{5} + 13309 T^{6} - 938 p T^{7} - 214 p^{2} T^{8} + 154 p^{3} T^{9} - 11 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 45 T^{2} + 72 T^{3} + 990 T^{4} - 1620 T^{5} - 20945 T^{6} - 1620 p T^{7} + 990 p^{2} T^{8} + 72 p^{3} T^{9} - 45 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 7 T - 17 T^{2} - 104 T^{3} + 461 T^{4} - 2231 T^{5} - 38666 T^{6} - 2231 p T^{7} + 461 p^{2} T^{8} - 104 p^{3} T^{9} - 17 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + T + 56 T^{2} + 133 T^{3} + 56 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 4 T + 79 T^{2} - 332 T^{3} + 79 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 10 T - 35 T^{2} - 242 T^{3} + 4730 T^{4} + 13570 T^{5} - 143867 T^{6} + 13570 p T^{7} + 4730 p^{2} T^{8} - 242 p^{3} T^{9} - 35 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 13 T + p T^{2} )^{3} \) |
| 47 | \( 1 - 6 T - 9 T^{2} + 930 T^{3} - 3690 T^{4} - 14874 T^{5} + 392947 T^{6} - 14874 p T^{7} - 3690 p^{2} T^{8} + 930 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 2 T - 119 T^{2} - 38 T^{3} + 8402 T^{4} + 7666 T^{5} - 504503 T^{6} + 7666 p T^{7} + 8402 p^{2} T^{8} - 38 p^{3} T^{9} - 119 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 19 T + 73 T^{2} + 586 T^{3} + 19535 T^{4} + 124027 T^{5} + 228670 T^{6} + 124027 p T^{7} + 19535 p^{2} T^{8} + 586 p^{3} T^{9} + 73 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 9 T - 45 T^{2} + 284 T^{3} + 1809 T^{4} + 15525 T^{5} - 313194 T^{6} + 15525 p T^{7} + 1809 p^{2} T^{8} + 284 p^{3} T^{9} - 45 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 10 T + 15 T^{2} + 426 T^{3} - 590 T^{4} - 50990 T^{5} - 286505 T^{6} - 50990 p T^{7} - 590 p^{2} T^{8} + 426 p^{3} T^{9} + 15 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 17 T + T^{2} + 34 T^{3} + 20591 T^{4} - 100385 T^{5} - 553826 T^{6} - 100385 p T^{7} + 20591 p^{2} T^{8} + 34 p^{3} T^{9} + p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 4 T - 127 T^{2} + 924 T^{3} + 7018 T^{4} - 44768 T^{5} - 240503 T^{6} - 44768 p T^{7} + 7018 p^{2} T^{8} + 924 p^{3} T^{9} - 127 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 7 T - 159 T^{2} - 654 T^{3} + 19447 T^{4} + 33079 T^{5} - 1624562 T^{6} + 33079 p T^{7} + 19447 p^{2} T^{8} - 654 p^{3} T^{9} - 159 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + p T^{2} )^{6} \) |
| 89 | \( 1 - 9 T - 189 T^{2} + 756 T^{3} + 34533 T^{4} - 68859 T^{5} - 3182114 T^{6} - 68859 p T^{7} + 34533 p^{2} T^{8} + 756 p^{3} T^{9} - 189 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 8 T - 67 T^{2} - 1800 T^{3} - 6938 T^{4} + 58504 T^{5} + 1586329 T^{6} + 58504 p T^{7} - 6938 p^{2} T^{8} - 1800 p^{3} T^{9} - 67 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.60297457239646193371860414196, −5.50047444795480605473615821646, −5.12527121961347585086115477722, −5.05564068325907021099691728858, −5.05335202002362261970458141249, −5.04132452468578451201922401357, −4.86865830399438948402640867262, −4.51513710424722095410160159672, −4.26807698214335605336871959369, −4.12867598070293127442270290456, −3.88156633781335282792008990787, −3.70281177899608783973360560580, −3.28831626653203967889496407979, −3.18964867804754293255362322908, −3.01978762615933085619447505044, −2.51900529330654304013893099245, −2.51440157163552321460994915680, −2.47146273661509135958980768859, −2.02994869980025784886523812867, −1.86444768550210045004125809371, −1.83654800738561650800935495607, −1.64414928021479718420897213953, −1.30032700899642871063281104695, −1.12174822512070078898066327636, −0.26882885662505451771965231000,
0.26882885662505451771965231000, 1.12174822512070078898066327636, 1.30032700899642871063281104695, 1.64414928021479718420897213953, 1.83654800738561650800935495607, 1.86444768550210045004125809371, 2.02994869980025784886523812867, 2.47146273661509135958980768859, 2.51440157163552321460994915680, 2.51900529330654304013893099245, 3.01978762615933085619447505044, 3.18964867804754293255362322908, 3.28831626653203967889496407979, 3.70281177899608783973360560580, 3.88156633781335282792008990787, 4.12867598070293127442270290456, 4.26807698214335605336871959369, 4.51513710424722095410160159672, 4.86865830399438948402640867262, 5.04132452468578451201922401357, 5.05335202002362261970458141249, 5.05564068325907021099691728858, 5.12527121961347585086115477722, 5.50047444795480605473615821646, 5.60297457239646193371860414196
Plot not available for L-functions of degree greater than 10.
