| L(s) = 1 | − 3-s + 2·4-s + 5·5-s + 7-s − 9·11-s − 2·12-s + 6·13-s − 5·15-s − 5·16-s − 9·17-s + 22·19-s + 10·20-s − 21-s − 11·23-s + 10·25-s + 2·28-s + 6·29-s + 7·31-s + 9·33-s + 5·35-s − 12·37-s − 6·39-s + 41-s − 4·43-s − 18·44-s − 15·47-s + 5·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 2.23·5-s + 0.377·7-s − 2.71·11-s − 0.577·12-s + 1.66·13-s − 1.29·15-s − 5/4·16-s − 2.18·17-s + 5.04·19-s + 2.23·20-s − 0.218·21-s − 2.29·23-s + 2·25-s + 0.377·28-s + 1.11·29-s + 1.25·31-s + 1.56·33-s + 0.845·35-s − 1.97·37-s − 0.960·39-s + 0.156·41-s − 0.609·43-s − 2.71·44-s − 2.18·47-s + 0.721·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.675119453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.675119453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 17 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 60 T^{3} + 61 T^{4} - 60 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 9 T + 14 T^{2} - 147 T^{3} - 1001 T^{4} - 147 p T^{5} + 14 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 11 T + 67 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 11 T + 38 T^{2} + 125 T^{3} + 821 T^{4} + 125 p T^{5} + 38 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 7 T + 38 T^{2} - 329 T^{3} + 2725 T^{4} - 329 p T^{5} + 38 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 12 T + 17 T^{2} - 360 T^{3} - 2879 T^{4} - 360 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - T - 10 T^{2} - 229 T^{3} + 1679 T^{4} - 229 p T^{5} - 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 15 T + 53 T^{2} + 15 T^{3} + 484 T^{4} + 15 p T^{5} + 53 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - T - 52 T^{2} + 105 T^{3} + 2651 T^{4} + 105 p T^{5} - 52 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 10 T + 101 T^{2} - 990 T^{3} + 10961 T^{4} - 990 p T^{5} + 101 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 20 T + 129 T^{2} + 520 T^{3} + 4001 T^{4} + 520 p T^{5} + 129 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 5 T - 27 T^{2} + 385 T^{3} + 6524 T^{4} + 385 p T^{5} - 27 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 7 T - 2 T^{2} - 569 T^{3} + 8925 T^{4} - 569 p T^{5} - 2 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 - 24 T + 177 T^{2} - 392 T^{3} + 225 T^{4} - 392 p T^{5} + 177 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 21 T + 223 T^{2} - 2625 T^{3} + 30136 T^{4} - 2625 p T^{5} + 223 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 16 T + 257 T^{2} + 2868 T^{3} + 35165 T^{4} + 2868 p T^{5} + 257 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 865 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.14797210802030311082796579918, −7.14463894800781613131640615160, −6.91744411806359716055596052847, −6.39701039959756536287318615919, −6.34745393906952280469454860692, −6.29338388451645236116348755665, −5.88899684053943245605554334183, −5.86887578601019669462475775603, −5.48603350027895258420344376082, −5.15237705227576180851474008095, −5.08829855249410590591445932343, −4.92544126136336758492385159175, −4.82266587045424680147148116961, −4.34949359551702993910242742085, −3.90278339876144726664942241811, −3.55325973405149713124380433055, −3.12473936808948237651006938590, −2.95197217021568788619982616367, −2.89390612638963273856600776176, −2.23908244494546858952363545632, −2.16322337321163602236814597847, −1.81867664188793009421331388960, −1.70192767795854775939820976562, −1.07087159161992430178554174663, −0.46572899625374888293672892222,
0.46572899625374888293672892222, 1.07087159161992430178554174663, 1.70192767795854775939820976562, 1.81867664188793009421331388960, 2.16322337321163602236814597847, 2.23908244494546858952363545632, 2.89390612638963273856600776176, 2.95197217021568788619982616367, 3.12473936808948237651006938590, 3.55325973405149713124380433055, 3.90278339876144726664942241811, 4.34949359551702993910242742085, 4.82266587045424680147148116961, 4.92544126136336758492385159175, 5.08829855249410590591445932343, 5.15237705227576180851474008095, 5.48603350027895258420344376082, 5.86887578601019669462475775603, 5.88899684053943245605554334183, 6.29338388451645236116348755665, 6.34745393906952280469454860692, 6.39701039959756536287318615919, 6.91744411806359716055596052847, 7.14463894800781613131640615160, 7.14797210802030311082796579918
