| L(s) = 1 | + 12·9-s − 6·11-s − 6·19-s + 6·29-s − 30·31-s − 18·41-s + 24·49-s − 12·59-s − 18·61-s − 18·71-s + 24·79-s + 72·81-s − 12·89-s − 72·99-s − 24·101-s + 66·109-s − 27·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 33·169-s + ⋯ |
| L(s) = 1 | + 4·9-s − 1.80·11-s − 1.37·19-s + 1.11·29-s − 5.38·31-s − 2.81·41-s + 24/7·49-s − 1.56·59-s − 2.30·61-s − 2.13·71-s + 2.70·79-s + 8·81-s − 1.27·89-s − 7.23·99-s − 2.38·101-s + 6.32·109-s − 2.45·121-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 + 2.53·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1021219475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1021219475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - 4 p T^{2} + 8 p^{2} T^{4} - 269 T^{6} + 8 p^{4} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 24 T^{2} + 312 T^{4} - 2621 T^{6} + 312 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 3 T + 27 T^{2} + 49 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 33 T^{2} + 417 T^{4} - 3793 T^{6} + 417 p^{2} T^{8} - 33 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 84 T^{2} + 3180 T^{4} - 69393 T^{6} + 3180 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 48 T^{2} + 1452 T^{4} - 32421 T^{6} + 1452 p^{2} T^{8} - 48 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 3 T + 51 T^{2} - 225 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 15 T + 159 T^{2} + 1019 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 81 T^{2} + 1233 T^{4} + 827 p T^{6} + 1233 p^{2} T^{8} - 81 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 231 T^{2} + 23286 T^{4} - 1307011 T^{6} + 23286 p^{2} T^{8} - 231 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 237 T^{2} + 24897 T^{4} - 1500609 T^{6} + 24897 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 219 T^{2} + 23838 T^{4} - 1581919 T^{6} + 23838 p^{2} T^{8} - 219 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 6 T + 105 T^{2} + 412 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 9 T + 117 T^{2} + 865 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 381 T^{2} + 61845 T^{4} - 5467849 T^{6} + 61845 p^{2} T^{8} - 381 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 9 T + 192 T^{2} + 1097 T^{3} + 192 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 393 T^{2} + 66957 T^{4} - 6365081 T^{6} + 66957 p^{2} T^{8} - 393 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 258 T^{2} - 1825 T^{3} + 258 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 213 T^{2} + 24321 T^{4} - 2162625 T^{6} + 24321 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 258 T^{2} + 1017 T^{3} + 258 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 321 T^{2} + 43473 T^{4} - 4207081 T^{6} + 43473 p^{2} T^{8} - 321 p^{4} T^{10} + p^{6} T^{12} \) |
| 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
−4.42115863555398423969183010761, −4.24516276357880618264776846238, −4.18556044066584892034250921701, −4.03646227151723195465242112445, −3.71819956131482776892507752241, −3.65071992631756074727208803121, −3.47444658216284777387716226444, −3.45748708349848142013525935853, −3.36571280805303564520261683484, −3.22035561157570939371629440643, −2.78471797770305866376345666209, −2.76552139917970842833926793925, −2.46660618482711564104880725569, −2.37141159328834063770073817856, −2.12171625357295156442403259898, −2.09747928391501751434868957842, −1.84908699327760012402440808353, −1.79172157845969179923588481772, −1.50601740205259182684811960351, −1.35528109140582765790626247289, −1.32219762162599696935788915247, −1.12102964089180988767236663900, −0.62440370078852320082160612569, −0.39252220876828987987954635959, −0.03559923574072089856798902916,
0.03559923574072089856798902916, 0.39252220876828987987954635959, 0.62440370078852320082160612569, 1.12102964089180988767236663900, 1.32219762162599696935788915247, 1.35528109140582765790626247289, 1.50601740205259182684811960351, 1.79172157845969179923588481772, 1.84908699327760012402440808353, 2.09747928391501751434868957842, 2.12171625357295156442403259898, 2.37141159328834063770073817856, 2.46660618482711564104880725569, 2.76552139917970842833926793925, 2.78471797770305866376345666209, 3.22035561157570939371629440643, 3.36571280805303564520261683484, 3.45748708349848142013525935853, 3.47444658216284777387716226444, 3.65071992631756074727208803121, 3.71819956131482776892507752241, 4.03646227151723195465242112445, 4.18556044066584892034250921701, 4.24516276357880618264776846238, 4.42115863555398423969183010761
Plot not available for L-functions of degree greater than 10.
