| L(s) = 1 | − 66·13-s + 33·19-s + 27·27-s − 66·43-s − 363·61-s − 64·64-s + 366·67-s + 429·73-s − 426·79-s − 363·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 5.07·13-s + 1.73·19-s + 27-s − 1.53·43-s − 5.95·61-s − 64-s + 5.46·67-s + 5.87·73-s − 5.39·79-s − 3·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 14.1·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1076033844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1076033844\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T^{3} + p^{6} T^{6} \) |
| 7 | \( 1 - 683 T^{3} + p^{6} T^{6} \) |
| 19 | \( ( 1 - 11 T + p^{2} T^{2} )^{3} \) |
| good | 2 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 5 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 11 | \( ( 1 - p T + p^{2} T^{2} )^{3}( 1 + p T + p^{2} T^{2} )^{3} \) |
| 13 | \( ( 1 + 22 T + p^{2} T^{2} )^{3}( 1 + 4033 T^{3} + p^{6} T^{6} ) \) |
| 17 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 23 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 29 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 31 | \( ( 1 - 23939 T^{3} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - 86183 T^{3} + p^{6} T^{6} )^{2} \) |
| 41 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 43 | \( ( 1 + 22 T + p^{2} T^{2} )^{3}( 1 + 153973 T^{3} + p^{6} T^{6} ) \) |
| 47 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 53 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 59 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 61 | \( ( 1 + 121 T + p^{2} T^{2} )^{3}( 1 - 62999 T^{3} + p^{6} T^{6} ) \) |
| 67 | \( ( 1 - 122 T + p^{2} T^{2} )^{3}( 1 - 412523 T^{3} + p^{6} T^{6} ) \) |
| 71 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 73 | \( ( 1 - 143 T + p^{2} T^{2} )^{3}( 1 + 704593 T^{3} + p^{6} T^{6} ) \) |
| 79 | \( ( 1 + 142 T + p^{2} T^{2} )^{3}( 1 - 937691 T^{3} + p^{6} T^{6} ) \) |
| 83 | \( ( 1 - p T + p^{2} T^{2} )^{3}( 1 + p T + p^{2} T^{2} )^{3} \) |
| 89 | \( ( 1 - p^{3} T^{3} + p^{6} T^{6} )( 1 + p^{3} T^{3} + p^{6} T^{6} ) \) |
| 97 | \( ( 1 - 1608263 T^{3} + p^{6} T^{6} )( 1 + 1551817 T^{3} + p^{6} T^{6} ) \) |
| 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
−5.74802673337417979135182544961, −5.60762204732192196573392019993, −5.53214726023978977968757556096, −5.20234139331908268204572121470, −4.99710078934168628416718502833, −4.93302310646322100980123903344, −4.92781149655435464289774232770, −4.63104458787772720615562156681, −4.59428519683148199066426558651, −4.29636470261854762346377198920, −4.00584780053260203348291388615, −3.68105514474847569759617126067, −3.53391724199935647126959604799, −3.35102638679011909386637835700, −2.97453980893957525623159245232, −2.84916865987902930525108344348, −2.68799210199418597071039448066, −2.41480147730649315635806417643, −2.24394756475003907257202575349, −2.12995363117709966172023371558, −1.56424454283147001641049918538, −1.34761652696394459809620870333, −1.04350365079803758625850747711, −0.43440139930809261831583487216, −0.06745742420881837709815904201,
0.06745742420881837709815904201, 0.43440139930809261831583487216, 1.04350365079803758625850747711, 1.34761652696394459809620870333, 1.56424454283147001641049918538, 2.12995363117709966172023371558, 2.24394756475003907257202575349, 2.41480147730649315635806417643, 2.68799210199418597071039448066, 2.84916865987902930525108344348, 2.97453980893957525623159245232, 3.35102638679011909386637835700, 3.53391724199935647126959604799, 3.68105514474847569759617126067, 4.00584780053260203348291388615, 4.29636470261854762346377198920, 4.59428519683148199066426558651, 4.63104458787772720615562156681, 4.92781149655435464289774232770, 4.93302310646322100980123903344, 4.99710078934168628416718502833, 5.20234139331908268204572121470, 5.53214726023978977968757556096, 5.60762204732192196573392019993, 5.74802673337417979135182544961
Plot not available for L-functions of degree greater than 10.
