| L(s) = 1 | + 2·7-s − 12·11-s − 4·23-s + 4·37-s + 40·43-s + 2·49-s − 40·53-s − 40·67-s + 8·71-s − 24·77-s + 3·81-s + 32·107-s + 76·113-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 3.61·11-s − 0.834·23-s + 0.657·37-s + 6.09·43-s + 2/7·49-s − 5.49·53-s − 4.88·67-s + 0.949·71-s − 2.73·77-s + 1/3·81-s + 3.09·107-s + 7.14·113-s + 3.09·121-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 − 0.630·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4672291613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4672291613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2 T + 2 T^{2} + 26 T^{3} - 62 T^{4} + 26 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 - p T^{4} - 92 T^{8} - p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 357 T^{4} + 68228 T^{8} + 357 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 + 69 T^{4} - 53260 T^{8} + 69 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 2 T + 2 T^{2} + 6 T^{3} - 382 T^{4} + 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 83 T^{2} + 3148 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 18 T^{2} + 1962 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + 2 T^{2} - 34 T^{3} + 178 T^{4} - 34 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 106 T^{2} + 6130 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 2227 T^{4} + 6943924 T^{8} - 2227 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 59 | \( ( 1 + 178 T^{2} + 14842 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T^{2} + 6822 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 14144 T^{4} + 103908670 T^{8} + 14144 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 271 T^{2} + 30340 T^{4} - 271 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 8400 T^{4} + 51732158 T^{8} - 8400 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 168 T^{2} + 14862 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 16837 T^{4} + 160234548 T^{8} + 16837 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.63391847473389389817894708508, −4.58023615794070133768660079490, −4.30532348117510951279050167625, −4.14650218819773019305120296947, −4.08203660962612651457157954382, −3.86865112457054119092447026393, −3.79445291350391243776887792972, −3.53823372381464758069731016411, −3.30021799926098113756272269906, −3.05857033331752064113336916462, −3.01618028801545408523662202322, −2.98330015600952953633873626733, −2.94641993904798019925147598790, −2.75404319294035051556935970805, −2.37249253522553260176745442357, −2.29143367228922813436735895524, −2.13080679224906015191774966487, −2.11504094815000047692519550964, −1.91217581114011036077437377240, −1.60560614553515516178816592482, −1.41042005595908317200264272396, −1.04205589522855756456871570279, −0.907129716014309692566401199852, −0.52940526039348851453800566659, −0.12106508757464322921694576946,
0.12106508757464322921694576946, 0.52940526039348851453800566659, 0.907129716014309692566401199852, 1.04205589522855756456871570279, 1.41042005595908317200264272396, 1.60560614553515516178816592482, 1.91217581114011036077437377240, 2.11504094815000047692519550964, 2.13080679224906015191774966487, 2.29143367228922813436735895524, 2.37249253522553260176745442357, 2.75404319294035051556935970805, 2.94641993904798019925147598790, 2.98330015600952953633873626733, 3.01618028801545408523662202322, 3.05857033331752064113336916462, 3.30021799926098113756272269906, 3.53823372381464758069731016411, 3.79445291350391243776887792972, 3.86865112457054119092447026393, 4.08203660962612651457157954382, 4.14650218819773019305120296947, 4.30532348117510951279050167625, 4.58023615794070133768660079490, 4.63391847473389389817894708508
Plot not available for L-functions of degree greater than 10.
