| L(s) = 1 | + 8·11-s − 8·19-s + 25-s − 16·31-s − 3·49-s − 48·59-s + 12·61-s + 8·71-s − 16·79-s − 48·89-s + 16·101-s + 20·109-s + 14·121-s + 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 30·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.83·19-s + 1/5·25-s − 2.87·31-s − 3/7·49-s − 6.24·59-s + 1.53·61-s + 0.949·71-s − 1.80·79-s − 5.08·89-s + 1.59·101-s + 1.91·109-s + 1.27·121-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2179183610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2179183610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 30 T^{2} + 359 T^{4} - 3332 T^{6} + 359 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 74 T^{2} + 2607 T^{4} - 55628 T^{6} + 2607 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 4 T + 49 T^{2} + 136 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 58 T^{2} + 1215 T^{4} - 18604 T^{6} + 1215 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 8 T + 53 T^{2} + 192 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - p T^{2} )^{6} \) |
| 41 | \( ( 1 + 107 T^{2} + 16 T^{3} + 107 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 130 T^{2} + 9815 T^{4} - 505980 T^{6} + 9815 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{3} \) |
| 53 | \( 1 + 14 T^{2} + 2327 T^{4} + 109668 T^{6} + 2327 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + p T^{2} )^{6} \) |
| 61 | \( ( 1 - 6 T + 131 T^{2} - 740 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 210 T^{2} + 20999 T^{4} - 1513628 T^{6} + 20999 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 4 T + 181 T^{2} - 600 T^{3} + 181 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 166 T^{2} + 19007 T^{4} - 1537812 T^{6} + 19007 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 8 T + 205 T^{2} + 1136 T^{3} + 205 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 370 T^{2} + 64935 T^{4} - 6808540 T^{6} + 64935 p^{2} T^{8} - 370 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 24 T + 443 T^{2} + 4640 T^{3} + 443 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 310 T^{2} + 54095 T^{4} - 6240180 T^{6} + 54095 p^{2} T^{8} - 310 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−4.44521900746137331845747998349, −4.42697285329493681694750795595, −4.33843185947599781286522845925, −4.14255303704429055479599420232, −4.12981121412760372149662522061, −3.86256849816596125127902866183, −3.69517849200429607151934409311, −3.60694818863090501378557370226, −3.52349191240472781691865209619, −3.26154032912988675556968880147, −3.02240108694921849847681017584, −3.01625290439400968323069937684, −2.91109030701819633854115115237, −2.54615015046123586613328801107, −2.33809789825244850828747042609, −2.11197654158810553189738600580, −2.00767211007748065978656218137, −1.90656967279179839841441905190, −1.70757891664267792099700110730, −1.32440715056381065272891318308, −1.23853556621950549128151904501, −1.20038993090343952174463594383, −1.00803599752532719023917951486, −0.19549673969457262917777200725, −0.10994208649135199512835570017,
0.10994208649135199512835570017, 0.19549673969457262917777200725, 1.00803599752532719023917951486, 1.20038993090343952174463594383, 1.23853556621950549128151904501, 1.32440715056381065272891318308, 1.70757891664267792099700110730, 1.90656967279179839841441905190, 2.00767211007748065978656218137, 2.11197654158810553189738600580, 2.33809789825244850828747042609, 2.54615015046123586613328801107, 2.91109030701819633854115115237, 3.01625290439400968323069937684, 3.02240108694921849847681017584, 3.26154032912988675556968880147, 3.52349191240472781691865209619, 3.60694818863090501378557370226, 3.69517849200429607151934409311, 3.86256849816596125127902866183, 4.12981121412760372149662522061, 4.14255303704429055479599420232, 4.33843185947599781286522845925, 4.42697285329493681694750795595, 4.44521900746137331845747998349
Plot not available for L-functions of degree greater than 10.
