| L(s) = 1 | + 2-s − 3·4-s + 3·5-s + 7-s − 5·8-s + 3·10-s + 11·11-s + 7·13-s + 14-s − 12·17-s − 10·19-s − 9·20-s + 11·22-s + 8·23-s + 10·25-s + 7·26-s − 3·28-s − 6·29-s − 12·31-s + 9·32-s − 12·34-s + 3·35-s + 9·37-s − 10·38-s − 15·40-s + 3·41-s − 33·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3/2·4-s + 1.34·5-s + 0.377·7-s − 1.76·8-s + 0.948·10-s + 3.31·11-s + 1.94·13-s + 0.267·14-s − 2.91·17-s − 2.29·19-s − 2.01·20-s + 2.34·22-s + 1.66·23-s + 2·25-s + 1.37·26-s − 0.566·28-s − 1.11·29-s − 2.15·31-s + 1.59·32-s − 2.05·34-s + 0.507·35-s + 1.47·37-s − 1.62·38-s − 2.37·40-s + 0.468·41-s − 4.97·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273121489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273121489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_4$ | \( 1 - p T + 51 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 3 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 49 p T^{5} + 6 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 63 T^{2} + 14 p T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + 63 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 307 p T^{5} - 6 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 3 T - 22 T^{2} - 171 T^{3} + 2215 T^{4} - 171 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 335 p T^{5} + 67 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 4 T - 47 T^{2} - 160 T^{3} + 2121 T^{4} - 160 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 6 T + 17 T^{2} + 222 T^{3} - 1685 T^{4} + 222 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 2559 p T^{5} + 245 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 15 T + 119 T^{2} + 1455 T^{3} + 17296 T^{4} + 1455 p T^{5} + 119 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 850 p T^{5} + 63 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 407 p T^{5} + 42 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 421 p T^{5} - 14 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 209 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 765 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−10.58914308488668776124766034680, −9.621233081045285182019871440769, −9.475806944679677370225573928854, −9.374689771408453280143777306937, −9.159105410768445855660326247727, −8.783123492788275939600923885725, −8.748218401930449856053425731779, −8.528606043985546045880646587795, −8.465909557683005444294389198268, −7.31212599076008857144553093446, −7.30266765817157788565595770418, −6.64230974815025429215943007626, −6.36893665835094932040845400721, −6.27743133691307765662514150910, −6.25377628042621661693132216485, −5.68557377996537153024881361506, −5.09379503607462220272508286657, −4.71167265570412505244281419165, −4.44893809068220476898388007449, −4.12309431720945309107310287530, −3.95878818678210987395774707797, −3.48286501579617997147708652733, −2.75797128229835304904206030614, −1.83112140241498648073084958194, −1.54716083591508772365809195642,
1.54716083591508772365809195642, 1.83112140241498648073084958194, 2.75797128229835304904206030614, 3.48286501579617997147708652733, 3.95878818678210987395774707797, 4.12309431720945309107310287530, 4.44893809068220476898388007449, 4.71167265570412505244281419165, 5.09379503607462220272508286657, 5.68557377996537153024881361506, 6.25377628042621661693132216485, 6.27743133691307765662514150910, 6.36893665835094932040845400721, 6.64230974815025429215943007626, 7.30266765817157788565595770418, 7.31212599076008857144553093446, 8.465909557683005444294389198268, 8.528606043985546045880646587795, 8.748218401930449856053425731779, 8.783123492788275939600923885725, 9.159105410768445855660326247727, 9.374689771408453280143777306937, 9.475806944679677370225573928854, 9.621233081045285182019871440769, 10.58914308488668776124766034680
