| L(s) = 1 | + 8·13-s + 32·25-s − 16·37-s + 8·49-s + 32·61-s − 32·73-s − 32·97-s + 8·109-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 32/5·25-s − 2.63·37-s + 8/7·49-s + 4.09·61-s − 3.74·73-s − 3.24·97-s + 0.766·109-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 − 1.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5980803797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5980803797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 16 T^{2} + 111 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T^{2} + 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 8 T^{2} + 6 p T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 40 T^{2} + 903 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T^{2} + 618 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 32 T^{2} + 546 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 88 T^{2} + 3543 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 52 T^{2} + 1830 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 75 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 88 T^{2} + 4530 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 148 T^{2} + 9162 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 140 T^{2} + 9270 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 188 T^{2} + 15750 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 244 T^{2} + 23850 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 176 T^{2} + 16098 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 196 T^{2} + 18618 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 188 T^{2} + 19542 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 160 T^{2} + 15039 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.26807872176677025339529006960, −3.26542085755795501321347958315, −3.09747628670580559840521298759, −3.03156989336735795545906120311, −3.01894209429748805524953379396, −2.81285492829708577028769959258, −2.74288980349910153308075295326, −2.55379006693510341173056722234, −2.45482195008399249063886657016, −2.44018506500183144216965536872, −2.24949131944379505686694395888, −2.16793889702403876024247838052, −1.85018195275273921278925071089, −1.73872165070198527161729450710, −1.57847406417961957438476853152, −1.55638780311269896514541596323, −1.50419721924123106474787812325, −1.33029211431380271398746108660, −1.02147561399620629105075894009, −1.01248842257613704474178393395, −0.870084507894451560915719434676, −0.797275706765099965441842950747, −0.63180075974528775138833177757, −0.41531188006269512539983873208, −0.03414105499779655982440219703,
0.03414105499779655982440219703, 0.41531188006269512539983873208, 0.63180075974528775138833177757, 0.797275706765099965441842950747, 0.870084507894451560915719434676, 1.01248842257613704474178393395, 1.02147561399620629105075894009, 1.33029211431380271398746108660, 1.50419721924123106474787812325, 1.55638780311269896514541596323, 1.57847406417961957438476853152, 1.73872165070198527161729450710, 1.85018195275273921278925071089, 2.16793889702403876024247838052, 2.24949131944379505686694395888, 2.44018506500183144216965536872, 2.45482195008399249063886657016, 2.55379006693510341173056722234, 2.74288980349910153308075295326, 2.81285492829708577028769959258, 3.01894209429748805524953379396, 3.03156989336735795545906120311, 3.09747628670580559840521298759, 3.26542085755795501321347958315, 3.26807872176677025339529006960
Plot not available for L-functions of degree greater than 10.
