| L(s) = 1 | − 2·4-s − 2·11-s − 3·16-s − 6·19-s − 36·29-s − 12·41-s + 4·44-s − 23·49-s − 20·59-s − 14·61-s + 8·64-s − 52·71-s + 12·76-s − 24·79-s − 24·89-s − 20·101-s + 40·109-s + 72·116-s − 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s − 0.603·11-s − 3/4·16-s − 1.37·19-s − 6.68·29-s − 1.87·41-s + 0.603·44-s − 3.28·49-s − 2.60·59-s − 1.79·61-s + 64-s − 6.17·71-s + 1.37·76-s − 2.70·79-s − 2.54·89-s − 1.99·101-s + 3.83·109-s + 6.68·116-s − 2.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 2 | \( 1 + p T^{2} + 7 T^{4} + 3 p^{2} T^{6} + 7 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 23 T^{2} + 307 T^{4} + 2586 T^{6} + 307 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + 17 T^{2} + 10 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 50 T^{2} + 1315 T^{4} + 21108 T^{6} + 1315 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 43 T^{2} + 1331 T^{4} + 25042 T^{6} + 1331 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 102 T^{2} + 4831 T^{4} + 138580 T^{6} + 4831 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 166 T^{2} + 13011 T^{4} + 608316 T^{6} + 13011 p^{2} T^{8} + 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 79 T^{2} + 516 T^{3} + 79 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 239 T^{2} + 24571 T^{4} + 1388154 T^{6} + 24571 p^{2} T^{8} + 239 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 95 T^{2} + 5443 T^{4} + 214314 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 162 T^{2} + 11539 T^{4} + 610708 T^{6} + 11539 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 10 T + 185 T^{2} + 1132 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 62 T^{2} + 4771 T^{4} + 199788 T^{6} + 4771 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 26 T + 413 T^{2} + 4124 T^{3} + 413 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 307 T^{2} + 43299 T^{4} + 3822498 T^{6} + 43299 p^{2} T^{8} + 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 229 T^{2} + 1864 T^{3} + 229 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 270 T^{2} + 39367 T^{4} + 3817060 T^{6} + 39367 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 554 T^{2} + 130507 T^{4} + 16717956 T^{6} + 130507 p^{2} T^{8} + 554 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.85899745658348104876768509492, −4.50646757123219558250114641473, −4.34089365526434628314804755903, −4.26080223924823850499479973217, −4.19906014668698273197896209512, −4.13579626074554309800000893590, −4.00046074058463134947503735020, −3.72496894954156675055169882685, −3.48132549283550199735641701122, −3.47870499295558170559892349945, −3.35957608609103798028880935546, −3.34525899521462470017835757745, −2.90226219681943277090419781772, −2.88145060430352573178639480945, −2.75650619466820822471354067945, −2.63095027650360229120305264862, −2.35837661969547337073244569692, −2.09665053626398941636759722964, −1.96381320794780510397011006791, −1.84716230080749488216895429778, −1.71060540388493841649297305056, −1.47420594182419079080223822244, −1.41591900276470370202155619893, −1.25336720337519930728368557478, −1.11349401184104800243792048397, 0, 0, 0, 0, 0, 0,
1.11349401184104800243792048397, 1.25336720337519930728368557478, 1.41591900276470370202155619893, 1.47420594182419079080223822244, 1.71060540388493841649297305056, 1.84716230080749488216895429778, 1.96381320794780510397011006791, 2.09665053626398941636759722964, 2.35837661969547337073244569692, 2.63095027650360229120305264862, 2.75650619466820822471354067945, 2.88145060430352573178639480945, 2.90226219681943277090419781772, 3.34525899521462470017835757745, 3.35957608609103798028880935546, 3.47870499295558170559892349945, 3.48132549283550199735641701122, 3.72496894954156675055169882685, 4.00046074058463134947503735020, 4.13579626074554309800000893590, 4.19906014668698273197896209512, 4.26080223924823850499479973217, 4.34089365526434628314804755903, 4.50646757123219558250114641473, 4.85899745658348104876768509492
Plot not available for L-functions of degree greater than 10.
