| L(s) = 1 | + 6·9-s − 16·17-s + 12·41-s − 16·49-s + 28·73-s + 27·81-s − 32·89-s + 36·97-s − 104·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·9-s − 3.88·17-s + 1.87·41-s − 2.28·49-s + 3.27·73-s + 3·81-s − 3.39·89-s + 3.65·97-s − 9.78·113-s + 4.72·121-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 − 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040384330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040384330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 14 T^{2} + 27 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 44 T^{2} + 1407 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 48 T^{2} + 1463 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 84 T^{2} + 4847 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 69 T^{2} + 1280 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 112 T^{2} + 8823 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 17 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 118 T^{2} + 7683 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.10904834142080680208218199135, −4.05671917098784115592497634714, −3.88074248135191052832206492813, −3.84804938759556932956895579481, −3.77430359607055028566675913111, −3.52820327954461375091435918424, −3.49840334045745115838349420947, −3.24629311627295587345124803465, −2.92920177501395108500403244880, −2.82711933683816202635238461747, −2.82659163759581521881084738684, −2.79096046752039208001417702146, −2.58502255568730084760684048240, −2.12591733517300823617626605424, −2.12456030084789774003236993866, −2.06719604266135215207457301960, −1.98379748561611631081146945806, −1.85401232129347677795160331513, −1.71447707169420221586987797073, −1.22799881117220384333217070861, −1.21968391557942937004785753262, −0.991161870433642469901843128828, −0.875380605229083321842357486432, −0.39176378277133973314657138967, −0.13203095838916446294954479009,
0.13203095838916446294954479009, 0.39176378277133973314657138967, 0.875380605229083321842357486432, 0.991161870433642469901843128828, 1.21968391557942937004785753262, 1.22799881117220384333217070861, 1.71447707169420221586987797073, 1.85401232129347677795160331513, 1.98379748561611631081146945806, 2.06719604266135215207457301960, 2.12456030084789774003236993866, 2.12591733517300823617626605424, 2.58502255568730084760684048240, 2.79096046752039208001417702146, 2.82659163759581521881084738684, 2.82711933683816202635238461747, 2.92920177501395108500403244880, 3.24629311627295587345124803465, 3.49840334045745115838349420947, 3.52820327954461375091435918424, 3.77430359607055028566675913111, 3.84804938759556932956895579481, 3.88074248135191052832206492813, 4.05671917098784115592497634714, 4.10904834142080680208218199135
Plot not available for L-functions of degree greater than 10.
