| L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s + 330·16-s + 16·23-s − 792·32-s − 128·46-s − 4·49-s + 16·53-s + 1.71e3·64-s − 16·79-s + 10·81-s + 576·92-s + 32·98-s − 128·106-s + 16·109-s − 48·113-s + 56·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·158-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s + 82.5·16-s + 3.33·23-s − 140.·32-s − 18.8·46-s − 4/7·49-s + 2.19·53-s + 214.5·64-s − 1.80·79-s + 10/9·81-s + 60.0·92-s + 3.23·98-s − 12.4·106-s + 1.53·109-s − 4.51·113-s + 5.09·121-s + 0.0887·127-s − 303.·128-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 + 10.1·158-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1224620570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1224620570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 - 10 T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 + 22 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 + 4 T^{2} - 10 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 13 | \( ( 1 + 40 T^{2} + 710 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 44 T^{2} + 950 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 24 T^{2} + 838 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 52 T^{2} + 1350 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 84 T^{2} + 3574 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 20 T^{2} + 38 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 84 T^{2} + 4678 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 44 T^{2} + 2390 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 108 T^{2} + 6886 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 216 T^{2} + 18598 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 136 T^{2} + 9798 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 140 T^{2} + 12086 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 28 T^{2} + 9270 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 252 T^{2} + 26422 T^{4} + 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 224 T^{2} + 26294 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 60 T^{2} + 14950 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 108 T^{2} + 16246 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−6.03062248722615198177416766796, −5.59034028837580024410420999398, −5.31438607244306327525566597832, −5.27955350781227689943737883212, −5.14364607542033502650341569336, −5.13951110018295177027555838325, −4.83685746274014639830070849504, −4.65619394929751210181300840883, −4.14657155283287519386544003128, −3.98818470394051712313603532686, −3.87001045777439211895733533279, −3.80786978680849867275871315589, −3.26538884246364051243265404772, −3.19099826398502361242395411892, −2.95887419098214329582836339777, −2.92475385206497316195899961231, −2.54850379386885800302515650190, −2.51723553906575536474786011466, −2.24690450593217753132228586265, −2.00707429774840263502805584996, −1.50797735151482452778993416098, −1.47576266495894743182816823569, −1.23833068979534546307491174632, −0.814560005156893192272222838560, −0.48365635517016648534400069353,
0.48365635517016648534400069353, 0.814560005156893192272222838560, 1.23833068979534546307491174632, 1.47576266495894743182816823569, 1.50797735151482452778993416098, 2.00707429774840263502805584996, 2.24690450593217753132228586265, 2.51723553906575536474786011466, 2.54850379386885800302515650190, 2.92475385206497316195899961231, 2.95887419098214329582836339777, 3.19099826398502361242395411892, 3.26538884246364051243265404772, 3.80786978680849867275871315589, 3.87001045777439211895733533279, 3.98818470394051712313603532686, 4.14657155283287519386544003128, 4.65619394929751210181300840883, 4.83685746274014639830070849504, 5.13951110018295177027555838325, 5.14364607542033502650341569336, 5.27955350781227689943737883212, 5.31438607244306327525566597832, 5.59034028837580024410420999398, 6.03062248722615198177416766796
Plot not available for L-functions of degree greater than 10.
