| L(s) = 1 | + 4-s + 10·9-s − 6·11-s − 2·16-s − 12·19-s + 20·29-s − 20·31-s + 10·36-s − 6·44-s − 3·49-s − 28·59-s + 20·61-s − 10·64-s − 48·71-s − 12·76-s − 16·79-s + 49·81-s − 40·89-s − 60·99-s + 16·101-s − 12·109-s + 20·116-s + 21·121-s − 20·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 10/3·9-s − 1.80·11-s − 1/2·16-s − 2.75·19-s + 3.71·29-s − 3.59·31-s + 5/3·36-s − 0.904·44-s − 3/7·49-s − 3.64·59-s + 2.56·61-s − 5/4·64-s − 5.69·71-s − 1.37·76-s − 1.80·79-s + 49/9·81-s − 4.23·89-s − 6.03·99-s + 1.59·101-s − 1.14·109-s + 1.85·116-s + 1.90·121-s − 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.275040397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275040397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 11 | \( ( 1 + T )^{6} \) |
| good | 2 | \( 1 - T^{2} + 3 T^{4} + 5 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - 10 T^{2} + 17 p T^{4} - 176 T^{6} + 17 p^{3} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 30 T^{2} + 515 T^{4} - 7640 T^{6} + 515 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 42 T^{2} + 1155 T^{4} - 24816 T^{6} + 1155 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 6 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 94 T^{2} + 4271 T^{4} - 120948 T^{6} + 4271 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 10 T + 99 T^{2} - 540 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 10 T + 113 T^{2} + 594 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 70 T^{2} + 3943 T^{4} - 191428 T^{6} + 3943 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 87 T^{2} - 54 T^{3} + 87 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 70 T^{2} + 2791 T^{4} - 68356 T^{6} + 2791 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 102 T^{2} + 5075 T^{4} - 209096 T^{6} + 5075 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 214 T^{2} + 20391 T^{4} - 1256932 T^{6} + 20391 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 14 T + 3 p T^{2} + 1578 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 123 T^{2} - 1282 T^{3} + 123 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 174 T^{2} + 19463 T^{4} - 1617812 T^{6} + 19463 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 24 T + 293 T^{2} + 2608 T^{3} + 293 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 106 T^{2} + 9475 T^{4} - 995536 T^{6} + 9475 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 8 T + 125 T^{2} + 1020 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 2 p T^{2} + 28487 T^{4} - 2384052 T^{6} + 28487 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 20 T + 315 T^{2} + 3240 T^{3} + 315 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 198 T^{2} + 29007 T^{4} - 3701140 T^{6} + 29007 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−4.61591867009673525662927064466, −4.57861386687208136392413157453, −4.43394815956325334041567838396, −4.37227504869229364848533794137, −4.22933656000982443989433708666, −4.21584355589939830284473724345, −4.10065722091876943732271427225, −3.66960262361683473590021063174, −3.56992958858925025792967828521, −3.51896121020202418661748651740, −3.04002041920816216941066277664, −2.97419947414149516084962456817, −2.82957248389742800511700881158, −2.67693356506046579204979815500, −2.61015906477758604299990354707, −2.50615228947633431354749355612, −1.91343608918237307081486210320, −1.76918009427112357994026879513, −1.70013175134360834801826608208, −1.64622089465333369012867550172, −1.58130009588248500882066060616, −1.34050116681965320086661755881, −0.65243444739390920322467575428, −0.58261001481125712865428619495, −0.19159736857780441853093724328,
0.19159736857780441853093724328, 0.58261001481125712865428619495, 0.65243444739390920322467575428, 1.34050116681965320086661755881, 1.58130009588248500882066060616, 1.64622089465333369012867550172, 1.70013175134360834801826608208, 1.76918009427112357994026879513, 1.91343608918237307081486210320, 2.50615228947633431354749355612, 2.61015906477758604299990354707, 2.67693356506046579204979815500, 2.82957248389742800511700881158, 2.97419947414149516084962456817, 3.04002041920816216941066277664, 3.51896121020202418661748651740, 3.56992958858925025792967828521, 3.66960262361683473590021063174, 4.10065722091876943732271427225, 4.21584355589939830284473724345, 4.22933656000982443989433708666, 4.37227504869229364848533794137, 4.43394815956325334041567838396, 4.57861386687208136392413157453, 4.61591867009673525662927064466
Plot not available for L-functions of degree greater than 10.
