| L(s) = 1 | + 8·4-s − 20·9-s + 40·16-s + 128·19-s − 256·31-s − 160·36-s − 28·49-s − 112·61-s + 160·64-s + 1.02e3·76-s + 256·79-s + 138·81-s + 144·109-s + 544·121-s − 2.04e3·124-s + 127-s + 131-s + 137-s + 139-s − 800·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 504·169-s − 2.56e3·171-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.22·9-s + 5/2·16-s + 6.73·19-s − 8.25·31-s − 4.44·36-s − 4/7·49-s − 1.83·61-s + 5/2·64-s + 13.4·76-s + 3.24·79-s + 1.70·81-s + 1.32·109-s + 4.49·121-s − 16.5·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.55·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.98·169-s − 14.9·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.282980418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.282980418\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} )^{4} \) |
| 3 | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 11 | \( ( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 252 T^{2} + 28198 T^{4} - 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 440 T^{2} + 204242 T^{4} + 440 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 16 T + p^{2} T^{2} )^{8} \) |
| 23 | \( ( 1 - 688 T^{2} + 666818 T^{4} - 688 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 64 T + 2246 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2338 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4580 T^{2} + 10468902 T^{4} - 4580 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4346 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 28 T + 4838 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 10532 T^{2} + 66420198 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 15932 T^{2} + 113802438 T^{4} - 15932 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 64 T + 10706 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 6356 T^{2} - 6983674 T^{4} + 6356 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 27832 T^{2} + 318232338 T^{4} - 27832 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 20540 T^{2} + 268016262 T^{4} - 20540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−3.84743393079751880094468931402, −3.64193846913668451350687605379, −3.55270267624104205177402583777, −3.49879008959662170751897412718, −3.43796523018642802539564882361, −3.39435429227886563006509776149, −3.33101752334210075991456680557, −3.03839985979313990361050405652, −2.99491618631417906690686458381, −2.91900417038703701652579305800, −2.76526458044318304276103213674, −2.39007065924449031750385939930, −2.24843877704961615214375485095, −2.19976585210789011193959103439, −2.09352788407123046263228675414, −2.08697526359971213669698900958, −1.53687802565022089856726828604, −1.50871916068412903754486393015, −1.44294068697220604739254303137, −1.34893278771388993003728161632, −0.982502729852538391478347781274, −0.961838516611115109215654122536, −0.56082995204069819894950510323, −0.26282220133297452365488722817, −0.19404651800842916816896183676,
0.19404651800842916816896183676, 0.26282220133297452365488722817, 0.56082995204069819894950510323, 0.961838516611115109215654122536, 0.982502729852538391478347781274, 1.34893278771388993003728161632, 1.44294068697220604739254303137, 1.50871916068412903754486393015, 1.53687802565022089856726828604, 2.08697526359971213669698900958, 2.09352788407123046263228675414, 2.19976585210789011193959103439, 2.24843877704961615214375485095, 2.39007065924449031750385939930, 2.76526458044318304276103213674, 2.91900417038703701652579305800, 2.99491618631417906690686458381, 3.03839985979313990361050405652, 3.33101752334210075991456680557, 3.39435429227886563006509776149, 3.43796523018642802539564882361, 3.49879008959662170751897412718, 3.55270267624104205177402583777, 3.64193846913668451350687605379, 3.84743393079751880094468931402
Plot not available for L-functions of degree greater than 10.
