| L(s) = 1 | + 3·4-s + 24·9-s + 4·16-s + 16·23-s + 40·25-s + 72·36-s + 12·37-s − 28·49-s − 20·53-s + 8·67-s + 32·71-s + 324·81-s + 48·92-s + 120·100-s − 4·113-s + 127-s + 131-s + 137-s + 139-s + 96·144-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 8·9-s + 16-s + 3.33·23-s + 8·25-s + 12·36-s + 1.97·37-s − 4·49-s − 2.74·53-s + 0.977·67-s + 3.79·71-s + 36·81-s + 5.00·92-s + 12·100-s − 0.376·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8·144-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(79.03614575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(79.03614575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + p T^{2} )^{4} \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( ( 1 - p T^{2} )^{8} \) |
| 5 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 144 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 228 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 696 p T^{5} + 101 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 696 p T^{5} + 101 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 60 p T^{5} + 47 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - p T^{2} )^{8} \) |
| 61 | \( ( 1 + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 472 p T^{5} - 51 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 1824 p T^{5} + 185 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 752 p T^{5} - 15 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 752 p T^{5} - 15 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−4.41467805800734792580816699415, −4.39296488393907645838339870708, −4.35237385799620989534818163165, −4.08706935041213194089878392114, −3.79895600880237954334450292572, −3.70152530341474903359205625575, −3.61349151147994840818710419649, −3.29387906797869271940522952365, −3.25869973863904166559423853896, −3.22356298032549640555120698204, −3.15187838387607303408813450081, −3.00541581884440210593320067391, −2.55156495935609882830578811439, −2.45006459327068992715057474973, −2.35771103081175849455557496045, −2.29798072493143830217384922643, −1.94863166678630413669948801857, −1.81641282920196295667708117055, −1.55533687789233353100234366481, −1.29532816596362424247493935780, −1.18253822086048807654132858204, −1.17914508720037327233992850196, −1.10932249005836802694084971313, −0.925205991849576659764398176401, −0.838784279811590194515384916481,
0.838784279811590194515384916481, 0.925205991849576659764398176401, 1.10932249005836802694084971313, 1.17914508720037327233992850196, 1.18253822086048807654132858204, 1.29532816596362424247493935780, 1.55533687789233353100234366481, 1.81641282920196295667708117055, 1.94863166678630413669948801857, 2.29798072493143830217384922643, 2.35771103081175849455557496045, 2.45006459327068992715057474973, 2.55156495935609882830578811439, 3.00541581884440210593320067391, 3.15187838387607303408813450081, 3.22356298032549640555120698204, 3.25869973863904166559423853896, 3.29387906797869271940522952365, 3.61349151147994840818710419649, 3.70152530341474903359205625575, 3.79895600880237954334450292572, 4.08706935041213194089878392114, 4.35237385799620989534818163165, 4.39296488393907645838339870708, 4.41467805800734792580816699415
Plot not available for L-functions of degree greater than 10.
