| L(s) = 1 | − 3·9-s − 12·11-s + 4·19-s − 5·25-s + 20·29-s + 24·31-s − 20·41-s + 42·49-s + 12·59-s + 4·61-s + 16·71-s − 40·79-s + 6·81-s + 44·89-s + 36·99-s − 60·101-s + 24·109-s + 38·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 9-s − 3.61·11-s + 0.917·19-s − 25-s + 3.71·29-s + 4.31·31-s − 3.12·41-s + 6·49-s + 1.56·59-s + 0.512·61-s + 1.89·71-s − 4.50·79-s + 2/3·81-s + 4.66·89-s + 3.61·99-s − 5.97·101-s + 2.29·109-s + 3.45·121-s − 0.715·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549785469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549785469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( ( 1 - p T^{2} )^{6} \) |
| 11 | \( ( 1 + 6 T + 35 T^{2} + 112 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 46 T^{2} + 1439 T^{4} - 28068 T^{6} + 1439 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 33 T^{2} - 92 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 54 T^{2} + 1871 T^{4} - 46868 T^{6} + 1871 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 10 T + 43 T^{2} - 108 T^{3} + 43 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 12 T + 101 T^{2} - 584 T^{3} + 101 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 130 T^{2} + 9335 T^{4} - 417660 T^{6} + 9335 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 10 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 202 T^{2} + 19015 T^{4} - 1043212 T^{6} + 19015 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 2 p T^{2} + 7139 T^{4} - 397884 T^{6} + 7139 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + 151 T^{2} - 212 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 130 T^{2} + 12871 T^{4} - 1093564 T^{6} + 12871 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 215 T^{2} - 1072 T^{3} + 215 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 106 T^{2} + 14783 T^{4} - 1127628 T^{6} + 14783 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 20 T + 345 T^{2} + 3320 T^{3} + 345 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 358 T^{2} + 62555 T^{4} - 6541356 T^{6} + 62555 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 22 T + 361 T^{2} - 3672 T^{3} + 361 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.11171818288867438888292609926, −4.78269923429936598028536343139, −4.64982929997061365432111977485, −4.63183174145057538038395255810, −4.36223527195295732273681589980, −4.10730708062395728568942631491, −3.94731023187876966275697102425, −3.91008759763318696290211277627, −3.87403364941065329325446591857, −3.25242509453202029788493913655, −3.23531791428257812801357020721, −3.05450190799556185236435560736, −2.92970900312235734574531812403, −2.68119735105710948267816707283, −2.64252810369303682756033193900, −2.62788764105150553599212179629, −2.32965637623613612762920136390, −2.22509669275629148811609855287, −2.02484172789241170997944818127, −1.61383289075840042996961143375, −1.23068546542277125054153530126, −1.02576401867867564325747385002, −0.73764770233626992639436083242, −0.67100685482534905698289186302, −0.25651970391370069379619951743,
0.25651970391370069379619951743, 0.67100685482534905698289186302, 0.73764770233626992639436083242, 1.02576401867867564325747385002, 1.23068546542277125054153530126, 1.61383289075840042996961143375, 2.02484172789241170997944818127, 2.22509669275629148811609855287, 2.32965637623613612762920136390, 2.62788764105150553599212179629, 2.64252810369303682756033193900, 2.68119735105710948267816707283, 2.92970900312235734574531812403, 3.05450190799556185236435560736, 3.23531791428257812801357020721, 3.25242509453202029788493913655, 3.87403364941065329325446591857, 3.91008759763318696290211277627, 3.94731023187876966275697102425, 4.10730708062395728568942631491, 4.36223527195295732273681589980, 4.63183174145057538038395255810, 4.64982929997061365432111977485, 4.78269923429936598028536343139, 5.11171818288867438888292609926
Plot not available for L-functions of degree greater than 10.
