| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s − 8·21-s − 2·23-s − 2·27-s + 6·29-s − 4·43-s − 4·47-s + 6·49-s − 2·61-s + 12·63-s + 4·69-s + 81-s + 2·83-s − 12·87-s − 8·101-s + 6·103-s − 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s − 12·147-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s − 8·21-s − 2·23-s − 2·27-s + 6·29-s − 4·43-s − 4·47-s + 6·49-s − 2·61-s + 12·63-s + 4·69-s + 81-s + 2·83-s − 12·87-s − 8·101-s + 6·103-s − 8·107-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + 8·141-s − 12·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.480445608\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480445608\times10^{-5}\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 13 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 23 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 37 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 41 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 47 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
| 53 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 71 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 73 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 79 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
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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
−3.72543614723940592243025582855, −3.70951708029968022373038195566, −3.40705796161011591993973174261, −3.36669611713169556130848060550, −3.31378896328819707873125047547, −3.12934380897232612678322026610, −2.93147685622055406538461647097, −2.81318755090589922217851250374, −2.71172869604517473426881947558, −2.60929555515067137449739993231, −2.56797105004540820397226871596, −2.43685760503564186696678873216, −2.24978459082322600216950942499, −1.97322287210696176740920862652, −1.80453810769698457888590891357, −1.78685198367493889744002326325, −1.77112715521194536998217613180, −1.49102229140475221554055949740, −1.32194223689711514608347402316, −1.29783510108590529105028596367, −1.21359031867850307218632711946, −1.21353372159341419248320091114, −1.04167940902770683982657533805, −0.67883757351304709131047255957, −0.000846644362155833575642995893,
0.000846644362155833575642995893, 0.67883757351304709131047255957, 1.04167940902770683982657533805, 1.21353372159341419248320091114, 1.21359031867850307218632711946, 1.29783510108590529105028596367, 1.32194223689711514608347402316, 1.49102229140475221554055949740, 1.77112715521194536998217613180, 1.78685198367493889744002326325, 1.80453810769698457888590891357, 1.97322287210696176740920862652, 2.24978459082322600216950942499, 2.43685760503564186696678873216, 2.56797105004540820397226871596, 2.60929555515067137449739993231, 2.71172869604517473426881947558, 2.81318755090589922217851250374, 2.93147685622055406538461647097, 3.12934380897232612678322026610, 3.31378896328819707873125047547, 3.36669611713169556130848060550, 3.40705796161011591993973174261, 3.70951708029968022373038195566, 3.72543614723940592243025582855
Plot not available for L-functions of degree greater than 10.
