| L(s) = 1 | + 6·3-s − 10·4-s + 3·9-s − 60·12-s − 105·16-s + 638·25-s + 54·27-s + 820·31-s − 30·36-s − 884·37-s − 630·48-s + 356·49-s + 1.56e3·64-s + 172·67-s + 3.82e3·75-s − 459·81-s + 4.92e3·93-s − 3.83e3·97-s − 6.38e3·100-s + 8.32e3·103-s − 540·108-s − 5.30e3·111-s − 872·121-s − 8.20e3·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 5/4·4-s + 1/9·9-s − 1.44·12-s − 1.64·16-s + 5.10·25-s + 0.384·27-s + 4.75·31-s − 0.138·36-s − 3.92·37-s − 1.89·48-s + 1.03·49-s + 3.04·64-s + 0.313·67-s + 5.89·75-s − 0.629·81-s + 5.48·93-s − 4.01·97-s − 6.37·100-s + 7.95·103-s − 0.481·108-s − 4.53·111-s − 0.655·121-s − 5.93·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.433733971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433733971\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - p T + 4 p T^{2} - p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | \( 1 + 872 T^{2} + 25230 p^{2} T^{4} + 872 p^{6} T^{6} + p^{12} T^{8} \) |
| good | 2 | \( ( 1 + 5 T^{2} + 45 p T^{4} + 5 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 5 | \( ( 1 - 319 T^{2} + 53106 T^{4} - 319 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 178 T^{2} + 94362 T^{4} - 178 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 7594 T^{2} + 23921970 T^{4} - 7594 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 3812 T^{2} + 51500166 T^{4} + 3812 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 + 3380 T^{2} - 20908650 T^{4} + 3380 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 47119 T^{2} + 851079294 T^{4} - 47119 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 + 84752 T^{2} + 2972083566 T^{4} + 84752 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 205 T + 69690 T^{2} - 205 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 + 221 T + 85860 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 + 148064 T^{2} + 13890568158 T^{4} + 148064 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 188068 T^{2} + 17913601926 T^{4} - 188068 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 258454 T^{2} + 703819350 p T^{4} - 258454 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 211354 T^{2} + 21017131314 T^{4} - 211354 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 709753 T^{2} + 209443663428 T^{4} - 709753 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 876394 T^{2} + 295021357554 T^{4} - 876394 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 43 T + 569730 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 - 823207 T^{2} + 410231814606 T^{4} - 823207 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 155524 T^{2} + 71302205670 T^{4} - 155524 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 984562 T^{2} + 499365566106 T^{4} - 984562 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 25972 p T^{2} + 1813793947254 T^{4} + 25972 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 801169 T^{2} + 974511119856 T^{4} - 801169 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 + 959 T + 1567410 T^{2} + 959 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.912877708061668856983498150440, −7.38046048772218343155171396972, −7.31888452655886280069096564305, −7.23057390285820135937755489295, −6.78024524585597075763815532555, −6.73426905145222999421611740115, −6.56183360819712869106845990079, −6.41729153359563786610934290608, −6.28504213408638895402795814910, −5.73823296033989153283522557238, −5.35863317535962362054223979642, −5.22037423818785453677103945367, −4.93641389081584091403408843223, −4.77784225500535470199415488227, −4.55066301893890813421406617855, −4.48160235690420026896603315875, −4.23110753246778158199509414786, −3.53464179954195476101783881965, −3.48191240059266032789049814932, −2.87941321607187626919338726394, −2.83146810916314971674465148256, −2.65798632735795558858850491929, −2.04747580549261023686850057389, −1.16633961322149226925673310597, −0.63875107001131564116439778891,
0.63875107001131564116439778891, 1.16633961322149226925673310597, 2.04747580549261023686850057389, 2.65798632735795558858850491929, 2.83146810916314971674465148256, 2.87941321607187626919338726394, 3.48191240059266032789049814932, 3.53464179954195476101783881965, 4.23110753246778158199509414786, 4.48160235690420026896603315875, 4.55066301893890813421406617855, 4.77784225500535470199415488227, 4.93641389081584091403408843223, 5.22037423818785453677103945367, 5.35863317535962362054223979642, 5.73823296033989153283522557238, 6.28504213408638895402795814910, 6.41729153359563786610934290608, 6.56183360819712869106845990079, 6.73426905145222999421611740115, 6.78024524585597075763815532555, 7.23057390285820135937755489295, 7.31888452655886280069096564305, 7.38046048772218343155171396972, 7.912877708061668856983498150440
Plot not available for L-functions of degree greater than 10.
