| L(s) = 1 | + 2·4-s + 4·7-s + 16-s + 8·25-s + 8·28-s − 16·37-s − 16·43-s + 18·49-s − 2·64-s − 32·67-s + 40·79-s + 16·100-s + 80·109-s + 4·112-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 40·169-s − 32·172-s + ⋯ |
| L(s) = 1 | + 4-s + 1.51·7-s + 1/4·16-s + 8/5·25-s + 1.51·28-s − 2.63·37-s − 2.43·43-s + 18/7·49-s − 1/4·64-s − 3.90·67-s + 4.50·79-s + 8/5·100-s + 7.66·109-s + 0.377·112-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.07·169-s − 2.43·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798109969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798109969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 22 T^{2} - 357 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 58 T^{2} + 1683 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 70 T^{2} + 2691 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 32 T^{2} - 2457 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 28 T^{2} - 2937 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 160 T^{2} + 18711 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + 170 T^{2} + 19491 T^{4} + 170 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.20803664220089564087435087609, −3.95458978231539740410451644877, −3.89481167934856389867900410432, −3.81277482729393081722216660919, −3.59456103539812716258165621470, −3.59103740701272334135111319919, −3.34598790844061005218052725199, −3.30666760498138646851810055640, −3.15631032968940887655490654387, −2.95289864226857084958278396510, −2.84508913463090600907026690123, −2.70302185799391594790098903518, −2.65084396659689621236777899250, −2.21959246135466579939384074822, −2.17068223303384374843705219824, −2.01573629991917285874962258723, −2.00410399304068148285485030774, −1.82595933138260372388140615981, −1.69524951540088176943988835088, −1.38737130034628442645727155079, −1.22386238742311741925763800420, −1.14431908871659870359626804471, −0.77468312068392539571631650601, −0.63274818063424262931442414244, −0.14411384948269391431524818713,
0.14411384948269391431524818713, 0.63274818063424262931442414244, 0.77468312068392539571631650601, 1.14431908871659870359626804471, 1.22386238742311741925763800420, 1.38737130034628442645727155079, 1.69524951540088176943988835088, 1.82595933138260372388140615981, 2.00410399304068148285485030774, 2.01573629991917285874962258723, 2.17068223303384374843705219824, 2.21959246135466579939384074822, 2.65084396659689621236777899250, 2.70302185799391594790098903518, 2.84508913463090600907026690123, 2.95289864226857084958278396510, 3.15631032968940887655490654387, 3.30666760498138646851810055640, 3.34598790844061005218052725199, 3.59103740701272334135111319919, 3.59456103539812716258165621470, 3.81277482729393081722216660919, 3.89481167934856389867900410432, 3.95458978231539740410451644877, 4.20803664220089564087435087609
Plot not available for L-functions of degree greater than 10.
