| L(s) = 1 | − 24·17-s + 8·29-s − 32·37-s + 24·41-s + 24·49-s + 64·97-s − 40·101-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.82·17-s + 1.48·29-s − 5.26·37-s + 3.74·41-s + 24/7·49-s + 6.49·97-s − 3.98·101-s − 1.09·121-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 + 5.53·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1134677801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1134677801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 12 T^{2} + 18 p T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 12 T^{2} + 114 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 12 T^{2} - 222 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 28 T^{2} + 934 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 12 T^{2} + 1638 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 108 T^{2} + 5634 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 52 T^{2} + 1174 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 128 T^{2} + 8434 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 180 T^{2} + 14982 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 28 T^{2} + 3718 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 60 T^{2} + 1878 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 180 T^{2} + 16182 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 268 T^{2} + 28534 T^{4} - 268 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 300 T^{2} + 34962 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 60 T^{2} - 1002 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 80 T^{2} + 1762 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.27728839053281319950026016814, −3.09052656345057729800228283834, −3.07106121194464277738174569271, −3.01512550286943919345457047382, −2.70164362831262705570292965601, −2.69234981066813287524978314074, −2.57910131223819410852251817429, −2.37739998517634033256243486947, −2.27737172444223649871619690968, −2.24045898431578007752262110741, −2.15210460816113704039598399570, −2.03737680998383219327705247699, −2.01883492830269149959977992307, −1.92745864124829816985089857774, −1.79697620963680959448450087350, −1.57983618096484909683600594343, −1.45389434747232904409217175391, −1.06160512900123811389989176312, −0.998267327725308075125694744810, −0.987907645398218270747670208900, −0.948105107857137971670927376151, −0.63501072256218107168617347696, −0.33851343854194891695492669925, −0.26606345660672156553428791596, −0.03645705830444903266975255947,
0.03645705830444903266975255947, 0.26606345660672156553428791596, 0.33851343854194891695492669925, 0.63501072256218107168617347696, 0.948105107857137971670927376151, 0.987907645398218270747670208900, 0.998267327725308075125694744810, 1.06160512900123811389989176312, 1.45389434747232904409217175391, 1.57983618096484909683600594343, 1.79697620963680959448450087350, 1.92745864124829816985089857774, 2.01883492830269149959977992307, 2.03737680998383219327705247699, 2.15210460816113704039598399570, 2.24045898431578007752262110741, 2.27737172444223649871619690968, 2.37739998517634033256243486947, 2.57910131223819410852251817429, 2.69234981066813287524978314074, 2.70164362831262705570292965601, 3.01512550286943919345457047382, 3.07106121194464277738174569271, 3.09052656345057729800228283834, 3.27728839053281319950026016814
Plot not available for L-functions of degree greater than 10.
