| L(s) = 1 | + 3-s − 5-s + 3·7-s − 9·11-s − 9·13-s − 15-s + 2·17-s + 10·19-s + 3·21-s + 4·23-s − 10·29-s − 8·31-s − 9·33-s − 3·35-s − 3·37-s − 9·39-s + 23·41-s − 16·43-s + 3·47-s + 7·49-s + 2·51-s + 6·53-s + 9·55-s + 10·57-s + 20·59-s + 3·61-s + 9·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2.71·11-s − 2.49·13-s − 0.258·15-s + 0.485·17-s + 2.29·19-s + 0.654·21-s + 0.834·23-s − 1.85·29-s − 1.43·31-s − 1.56·33-s − 0.507·35-s − 0.493·37-s − 1.44·39-s + 3.59·41-s − 2.43·43-s + 0.437·47-s + 49-s + 0.280·51-s + 0.824·53-s + 1.21·55-s + 1.32·57-s + 2.60·59-s + 0.384·61-s + 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656673231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656673231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 115 p T^{5} + 18 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 8 T + 3 T^{2} + 46 T^{3} + 1175 T^{4} + 46 p T^{5} + 3 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 961 p T^{5} + 208 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 3 T - 43 T^{2} + 45 T^{3} + 2116 T^{4} + 45 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 20 T + 131 T^{2} - 530 T^{3} + 3851 T^{4} - 530 p T^{5} + 131 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 441 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 27 T + 253 T^{2} - 819 T^{3} + 100 T^{4} - 819 p T^{5} + 253 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 130 p T^{5} - 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 5 T + 6 T^{2} + 715 T^{3} + 9821 T^{4} + 715 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 915 p T^{5} + 88 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 6655 p T^{5} + 537 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70524499128901909045963374440, −7.61406963808823430139356993047, −7.36148890830955698500590822973, −7.34895586975194386143716400010, −7.18210271064893032184712612093, −6.73843283263469067819251960737, −6.65276082638539208034085550126, −5.75763102117051697649734095292, −5.70189199944470387150408722903, −5.51443783545266190099106949633, −5.26459553189847253737609243245, −5.14577892442151859101857090510, −5.02280200566976835936531437414, −4.72919484341750827027418657608, −4.21339144885266706588915706933, −3.89224251299620582378531038708, −3.79258889268835168180548625901, −3.25679001194425205530546324413, −2.89973165096597087561672339159, −2.71434543541709152683857067768, −2.50200943325781693419355793934, −2.00893632880643572945802154640, −1.90386278304849756031703084614, −1.00736001656009906182059920955, −0.43860062469428206911064131805,
0.43860062469428206911064131805, 1.00736001656009906182059920955, 1.90386278304849756031703084614, 2.00893632880643572945802154640, 2.50200943325781693419355793934, 2.71434543541709152683857067768, 2.89973165096597087561672339159, 3.25679001194425205530546324413, 3.79258889268835168180548625901, 3.89224251299620582378531038708, 4.21339144885266706588915706933, 4.72919484341750827027418657608, 5.02280200566976835936531437414, 5.14577892442151859101857090510, 5.26459553189847253737609243245, 5.51443783545266190099106949633, 5.70189199944470387150408722903, 5.75763102117051697649734095292, 6.65276082638539208034085550126, 6.73843283263469067819251960737, 7.18210271064893032184712612093, 7.34895586975194386143716400010, 7.36148890830955698500590822973, 7.61406963808823430139356993047, 7.70524499128901909045963374440
