| L(s) = 1 | − 2·2-s + 4-s + 4·7-s − 2·8-s − 8·14-s + 7·16-s − 12·17-s + 8·23-s − 3·25-s + 4·28-s − 12·31-s − 10·32-s + 24·34-s + 20·41-s − 16·46-s − 8·47-s + 2·49-s + 6·50-s − 8·56-s + 24·62-s + 13·64-s − 12·68-s + 8·71-s − 36·73-s + 36·79-s − 40·82-s + 28·89-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.707·8-s − 2.13·14-s + 7/4·16-s − 2.91·17-s + 1.66·23-s − 3/5·25-s + 0.755·28-s − 2.15·31-s − 1.76·32-s + 4.11·34-s + 3.12·41-s − 2.35·46-s − 1.16·47-s + 2/7·49-s + 0.848·50-s − 1.06·56-s + 3.04·62-s + 13/8·64-s − 1.45·68-s + 0.949·71-s − 4.21·73-s + 4.05·79-s − 4.41·82-s + 2.96·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7399053370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7399053370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + 3 T^{2} + 3 p T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( ( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 407 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 6 T + 35 T^{2} + 172 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 57 T^{2} - 168 T^{3} + 57 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 6055 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 8087 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \) |
| 71 | \( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 37415 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−6.34036776782112605285565775844, −5.94528718389750326083641950583, −5.92755438926476484713544937907, −5.71309581233119933272279142852, −5.64019851830828581902514126793, −5.25391759228129386195291215817, −4.93638819227193315915037266741, −4.89949780802979217380907135256, −4.82175164523713210286754866229, −4.78515084122443021577770588885, −4.24194448948720143536838952211, −4.05416027035786545037430141463, −4.02794203578242678552985584258, −3.73585456244142742135525560450, −3.47604469765508479653182396190, −3.27941131882669984386011867813, −2.80427027647620278757369386090, −2.75100148090233691339341305142, −2.49642461781426211753342247840, −2.05213036832515664100053995864, −1.92238211221686359444567166570, −1.74712802993781640828135483907, −1.33880697152159371453870731140, −0.72295684515443759611612403378, −0.44325387913885196675219796241,
0.44325387913885196675219796241, 0.72295684515443759611612403378, 1.33880697152159371453870731140, 1.74712802993781640828135483907, 1.92238211221686359444567166570, 2.05213036832515664100053995864, 2.49642461781426211753342247840, 2.75100148090233691339341305142, 2.80427027647620278757369386090, 3.27941131882669984386011867813, 3.47604469765508479653182396190, 3.73585456244142742135525560450, 4.02794203578242678552985584258, 4.05416027035786545037430141463, 4.24194448948720143536838952211, 4.78515084122443021577770588885, 4.82175164523713210286754866229, 4.89949780802979217380907135256, 4.93638819227193315915037266741, 5.25391759228129386195291215817, 5.64019851830828581902514126793, 5.71309581233119933272279142852, 5.92755438926476484713544937907, 5.94528718389750326083641950583, 6.34036776782112605285565775844
Plot not available for L-functions of degree greater than 10.
