| L(s) = 1 | + 24·25-s − 16·53-s − 96·109-s − 64·113-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 24/5·25-s − 2.19·53-s − 9.19·109-s − 6.02·113-s + 1.45·121-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 + 1.84·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.355589581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.355589581\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 8 T^{2} - 30 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 12 T^{2} + 366 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 16 T^{2} + 768 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 96 T^{2} + 5568 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 136 T^{2} + 8034 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 52 T^{2} + 4806 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 224 T^{2} + 19488 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 108 T^{2} + 9966 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 124 T^{2} + 8214 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 192 T^{2} + 18624 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 172 T^{2} + 15270 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 320 T^{2} + 39360 T^{4} + 320 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 336 T^{2} + 44064 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 37632 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.13220426112256647424618900483, −3.12880023024701946366833850304, −3.07235273137101778782080456929, −2.80245825628599380101252443940, −2.74181521361410032376433681809, −2.71104034479886206260681781208, −2.60466287977132123773061771294, −2.57636597780909555170864142258, −2.49362129157439535727931787830, −2.29223623052115024147513713428, −2.03761809711855672360689665389, −1.98751764822370728065124761480, −1.84841153961151030834005007125, −1.59780052158720933297424871448, −1.56637619389729543880931822378, −1.50552461721036880474637338596, −1.46047863090228923679145190928, −1.19953723715482436785756879589, −1.09397424954775721763073756038, −1.00903206293473612293389700438, −0.77330672686462409095206322504, −0.70952354500150342783458480411, −0.42750203352663518727988414065, −0.29571910386667933406790353635, −0.20867801681667036833995013224,
0.20867801681667036833995013224, 0.29571910386667933406790353635, 0.42750203352663518727988414065, 0.70952354500150342783458480411, 0.77330672686462409095206322504, 1.00903206293473612293389700438, 1.09397424954775721763073756038, 1.19953723715482436785756879589, 1.46047863090228923679145190928, 1.50552461721036880474637338596, 1.56637619389729543880931822378, 1.59780052158720933297424871448, 1.84841153961151030834005007125, 1.98751764822370728065124761480, 2.03761809711855672360689665389, 2.29223623052115024147513713428, 2.49362129157439535727931787830, 2.57636597780909555170864142258, 2.60466287977132123773061771294, 2.71104034479886206260681781208, 2.74181521361410032376433681809, 2.80245825628599380101252443940, 3.07235273137101778782080456929, 3.12880023024701946366833850304, 3.13220426112256647424618900483
Plot not available for L-functions of degree greater than 10.
