| L(s) = 1 | + 2·4-s + 2·9-s − 12·11-s + 16-s − 8·29-s + 48·31-s + 4·36-s − 24·44-s − 8·49-s − 24·59-s − 20·61-s − 2·64-s + 44·71-s − 48·79-s + 81-s − 16·89-s − 24·99-s + 16·101-s + 8·109-s − 16·116-s + 98·121-s + 96·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | + 4-s + 2/3·9-s − 3.61·11-s + 1/4·16-s − 1.48·29-s + 8.62·31-s + 2/3·36-s − 3.61·44-s − 8/7·49-s − 3.12·59-s − 2.56·61-s − 1/4·64-s + 5.22·71-s − 5.40·79-s + 1/9·81-s − 1.69·89-s − 2.41·99-s + 1.59·101-s + 0.766·109-s − 1.48·116-s + 8.90·121-s + 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1234303141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1234303141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 14 T^{2} + 27 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 + 4 T^{2} - 33 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 + 16 T^{2} + 254 T^{4} - 9216 T^{6} - 157501 T^{8} - 9216 p^{2} T^{10} + 254 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 28 T^{2} + 423 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 70 T^{2} + 2657 T^{4} + 82950 T^{6} + 2159108 T^{8} + 82950 p^{2} T^{10} + 2657 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 4 T - 6 T^{2} - 144 T^{3} - 821 T^{4} - 144 p T^{5} - 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 - 34 T^{2} - 1511 T^{4} + 2414 T^{6} + 3656164 T^{8} + 2414 p^{2} T^{10} - 1511 p^{4} T^{12} - 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 8 T^{2} - 1617 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 144 T^{2} + 12014 T^{4} + 723456 T^{6} + 34313619 T^{8} + 723456 p^{2} T^{10} + 12014 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 160 T^{2} + 11378 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 10 T + 43 T^{2} - 650 T^{3} - 6572 T^{4} - 650 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 22 T + 231 T^{2} - 2442 T^{3} + 24604 T^{4} - 2442 p T^{5} + 231 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 162 T^{2} + 13219 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 10 T^{2} + 13163 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 8 T - 120 T^{2} + 48 T^{3} + 20239 T^{4} + 48 p T^{5} - 120 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 66 T^{2} + 3809 T^{4} - 1205886 T^{6} - 129215676 T^{8} - 1205886 p^{2} T^{10} + 3809 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.71138622993894337879499529405, −3.63182615586889566831606207354, −3.59205678070229762975642628734, −3.49014746206670299280451517858, −3.43790482327564088230284975164, −2.99884242443314952133893092664, −2.96089064787993921546580617378, −2.87202269873938630015122116637, −2.79495093576095436910024053678, −2.76621619774219096351730718806, −2.65777626449106021078551340781, −2.60729152937320153518753359152, −2.37927003529498807738094578383, −2.25998591556192755827862176041, −2.09222947394439593141247025137, −1.83635329930320886194402936645, −1.78262166114189413941131189005, −1.62948845732596183606672771204, −1.48389659182803053037959182007, −1.05863154599423683543359176453, −1.01268551122125552924945014874, −0.900136242030444701118745257303, −0.875085152751153338471966518835, −0.27343008915574141902863494950, −0.04144562858573414914770073029,
0.04144562858573414914770073029, 0.27343008915574141902863494950, 0.875085152751153338471966518835, 0.900136242030444701118745257303, 1.01268551122125552924945014874, 1.05863154599423683543359176453, 1.48389659182803053037959182007, 1.62948845732596183606672771204, 1.78262166114189413941131189005, 1.83635329930320886194402936645, 2.09222947394439593141247025137, 2.25998591556192755827862176041, 2.37927003529498807738094578383, 2.60729152937320153518753359152, 2.65777626449106021078551340781, 2.76621619774219096351730718806, 2.79495093576095436910024053678, 2.87202269873938630015122116637, 2.96089064787993921546580617378, 2.99884242443314952133893092664, 3.43790482327564088230284975164, 3.49014746206670299280451517858, 3.59205678070229762975642628734, 3.63182615586889566831606207354, 3.71138622993894337879499529405
Plot not available for L-functions of degree greater than 10.
