| L(s) = 1 | + 12·7-s + 16·13-s − 12·37-s + 20·43-s + 78·49-s − 32·61-s − 44·67-s + 192·91-s − 56·97-s − 48·103-s + 12·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 4.43·13-s − 1.97·37-s + 3.04·43-s + 78/7·49-s − 4.09·61-s − 5.37·67-s + 20.1·91-s − 5.68·97-s − 4.72·103-s + 1.14·109-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 + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.48563021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.48563021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - T )^{12} \) |
| good | 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{3}( 1 + 8 T^{2} + p^{2} T^{4} )^{3} \) |
| 11 | \( 1 + 2 p T^{4} + 1277 p T^{8} + 1317556 T^{12} + 1277 p^{5} T^{16} + 2 p^{9} T^{20} + p^{12} T^{24} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 88 T^{3} + 315 T^{4} - 1728 T^{5} + 7616 T^{6} - 1728 p T^{7} + 315 p^{2} T^{8} - 88 p^{3} T^{9} + 32 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 2 T^{2} + 223 T^{4} - 3076 T^{6} + 223 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + p^{2} T^{4} )^{6} \) |
| 23 | \( ( 1 - 108 T^{2} + 5363 T^{4} - 156760 T^{6} + 5363 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( 1 - 3530 T^{4} + 5536831 T^{8} - 5509074188 T^{12} + 5536831 p^{4} T^{16} - 3530 p^{8} T^{20} + p^{12} T^{24} \) |
| 31 | \( ( 1 - 106 T^{2} + 5071 T^{4} - 169228 T^{6} + 5071 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{6} \) |
| 41 | \( ( 1 + 118 T^{2} + 119 p T^{4} + 140692 T^{6} + 119 p^{3} T^{8} + 118 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 10 T + 50 T^{2} - 398 T^{3} - 1785 T^{4} + 35220 T^{5} - 183748 T^{6} + 35220 p T^{7} - 1785 p^{2} T^{8} - 398 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 154 T^{2} + 13423 T^{4} + 756268 T^{6} + 13423 p^{2} T^{8} + 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( 1 + 790 T^{4} - 2418209 T^{8} + 5899921588 T^{12} - 2418209 p^{4} T^{16} + 790 p^{8} T^{20} + p^{12} T^{24} \) |
| 59 | \( 1 - 6314 T^{4} + 48612511 T^{8} - 159936163532 T^{12} + 48612511 p^{4} T^{16} - 6314 p^{8} T^{20} + p^{12} T^{24} \) |
| 61 | \( ( 1 + 16 T + 128 T^{2} + 1440 T^{3} + 18779 T^{4} + 143024 T^{5} + 921472 T^{6} + 143024 p T^{7} + 18779 p^{2} T^{8} + 1440 p^{3} T^{9} + 128 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 22 T + 242 T^{2} + 2850 T^{3} + 35511 T^{4} + 315092 T^{5} + 2399612 T^{6} + 315092 p T^{7} + 35511 p^{2} T^{8} + 2850 p^{3} T^{9} + 242 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 188 T^{2} + 25939 T^{4} - 2091512 T^{6} + 25939 p^{2} T^{8} - 188 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 282 T^{2} + 42047 T^{4} - 3793004 T^{6} + 42047 p^{2} T^{8} - 282 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 154 T^{2} + 1711 T^{4} + 731348 T^{6} + 1711 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 12810 T^{4} + 46034687 T^{8} + 36989567092 T^{12} + 46034687 p^{4} T^{16} - 12810 p^{8} T^{20} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 46 T^{2} + 3343 T^{4} + 916708 T^{6} + 3343 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 14 T - 33 T^{2} - 1948 T^{3} - 33 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.53662381833418838916565159888, −2.37598121375847642995055827144, −2.30597280907186412651803080779, −2.27395290560899950717877051125, −2.18326882655671578243834125359, −2.09135353004027991184080017082, −1.94301065124882894089007013742, −1.79094298529089141311874978066, −1.69829262432842716030304100617, −1.62660643205833063534009967752, −1.58551218791754511876558404843, −1.57263441251176740726629978505, −1.53477961348518218953515954681, −1.48948620040732756206934720785, −1.28996803532622112715898430686, −1.21665449077925040764236874292, −1.14891882500326072162677785212, −1.13099388196479686736547314742, −1.00671678273470502183618657189, −0.905320382669864793567325801441, −0.800350933295364142105159723779, −0.55696138102526987282621870570, −0.34548316793586413472448547926, −0.20955324790804672774268000489, −0.10846412090085402637824905712,
0.10846412090085402637824905712, 0.20955324790804672774268000489, 0.34548316793586413472448547926, 0.55696138102526987282621870570, 0.800350933295364142105159723779, 0.905320382669864793567325801441, 1.00671678273470502183618657189, 1.13099388196479686736547314742, 1.14891882500326072162677785212, 1.21665449077925040764236874292, 1.28996803532622112715898430686, 1.48948620040732756206934720785, 1.53477961348518218953515954681, 1.57263441251176740726629978505, 1.58551218791754511876558404843, 1.62660643205833063534009967752, 1.69829262432842716030304100617, 1.79094298529089141311874978066, 1.94301065124882894089007013742, 2.09135353004027991184080017082, 2.18326882655671578243834125359, 2.27395290560899950717877051125, 2.30597280907186412651803080779, 2.37598121375847642995055827144, 2.53662381833418838916565159888
Plot not available for L-functions of degree greater than 10.
