| L(s) = 1 | − 4·3-s + 3·4-s + 6·9-s − 12·12-s + 16-s − 8·17-s − 20·23-s + 20·25-s − 4·27-s + 48·29-s + 18·36-s − 20·43-s − 4·48-s − 3·49-s + 32·51-s + 16·53-s − 12·61-s − 64-s − 24·68-s + 80·69-s − 80·75-s − 28·79-s − 7·81-s − 192·87-s − 60·92-s + 60·100-s − 64·101-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3/2·4-s + 2·9-s − 3.46·12-s + 1/4·16-s − 1.94·17-s − 4.17·23-s + 4·25-s − 0.769·27-s + 8.91·29-s + 3·36-s − 3.04·43-s − 0.577·48-s − 3/7·49-s + 4.48·51-s + 2.19·53-s − 1.53·61-s − 1/8·64-s − 2.91·68-s + 9.63·69-s − 9.23·75-s − 3.15·79-s − 7/9·81-s − 20.5·87-s − 6.25·92-s + 6·100-s − 6.36·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.071492189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071492189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} + p^{3} T^{4} - 5 p^{2} T^{6} + p^{5} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 4 p T^{2} + 192 T^{4} - 1166 T^{6} + 192 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 50 T^{2} + 1179 T^{4} - 16436 T^{6} + 1179 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 4 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 100 T^{2} + 4384 T^{4} - 108094 T^{6} + 4384 p^{2} T^{8} - 100 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 10 T + 70 T^{2} + 324 T^{3} + 70 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 132 T^{2} + 7952 T^{4} - 298646 T^{6} + 7952 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 106 T^{2} + 6731 T^{4} - 293796 T^{6} + 6731 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 186 T^{2} + 16127 T^{4} - 834860 T^{6} + 16127 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 10 T + 58 T^{2} + 232 T^{3} + 58 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 60 T^{2} + 6344 T^{4} - 242966 T^{6} + 6344 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 26 T^{2} + 4647 T^{4} - 304748 T^{6} + 4647 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 67 | \( 1 - 10 T^{2} + 1079 T^{4} + 296244 T^{6} + 1079 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 346 T^{2} + 53187 T^{4} - 4779748 T^{6} + 53187 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 140 T^{2} + 8200 T^{4} - 408038 T^{6} + 8200 p^{2} T^{8} - 140 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 188 T^{2} + 27456 T^{4} + 2388290 T^{6} + 27456 p^{2} T^{8} + 188 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 340 T^{2} + 60464 T^{4} - 6608166 T^{6} + 60464 p^{2} T^{8} - 340 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 540 T^{2} + 125240 T^{4} - 15959702 T^{6} + 125240 p^{2} T^{8} - 540 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.98676966003366157931403011100, −4.95055087164494135194294503703, −4.76189370129714084408958254134, −4.69361724017447109461350809707, −4.63112652596619947596383855799, −4.51814284840936683721311782164, −4.34016419538230622065122289447, −4.28484594528237478590613509647, −3.96779791187698894360331462799, −3.58693840095587296207611391122, −3.41974913888711034207875848598, −3.25501478348445739922291232627, −3.13241783977242895236028879296, −2.72385116470159023282838305025, −2.61487885678144888221684040194, −2.59969892386471425594724065270, −2.41668334275630003509019209442, −2.37834584677918873252825072152, −1.84189721348256383154906182178, −1.68406730606267219456071807991, −1.26392883376155372841798869972, −1.23147116243437430192303688059, −0.948929377273701275357502308683, −0.53283873545571116694495164634, −0.23376255480515898519890842593,
0.23376255480515898519890842593, 0.53283873545571116694495164634, 0.948929377273701275357502308683, 1.23147116243437430192303688059, 1.26392883376155372841798869972, 1.68406730606267219456071807991, 1.84189721348256383154906182178, 2.37834584677918873252825072152, 2.41668334275630003509019209442, 2.59969892386471425594724065270, 2.61487885678144888221684040194, 2.72385116470159023282838305025, 3.13241783977242895236028879296, 3.25501478348445739922291232627, 3.41974913888711034207875848598, 3.58693840095587296207611391122, 3.96779791187698894360331462799, 4.28484594528237478590613509647, 4.34016419538230622065122289447, 4.51814284840936683721311782164, 4.63112652596619947596383855799, 4.69361724017447109461350809707, 4.76189370129714084408958254134, 4.95055087164494135194294503703, 4.98676966003366157931403011100
Plot not available for L-functions of degree greater than 10.
