| L(s) = 1 | + 2·3-s − 4-s + 2·7-s + 6·11-s − 2·12-s − 3·16-s − 12·17-s + 4·19-s + 4·21-s + 6·23-s − 7·25-s − 2·27-s − 2·28-s − 6·29-s − 2·31-s + 12·33-s − 14·37-s + 10·43-s − 6·44-s + 12·47-s − 6·48-s + 3·49-s − 24·51-s − 6·53-s + 8·57-s + 18·59-s − 20·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.755·7-s + 1.80·11-s − 0.577·12-s − 3/4·16-s − 2.91·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s − 0.377·28-s − 1.11·29-s − 0.359·31-s + 2.08·33-s − 2.30·37-s + 1.52·43-s − 0.904·44-s + 1.75·47-s − 0.866·48-s + 3/7·49-s − 3.36·51-s − 0.824·53-s + 1.05·57-s + 2.34·59-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.667171459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.667171459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 195 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.419090959450620177767908620279, −9.324299252313651527737534493362, −9.117789009467200424218151363662, −9.026045909869793674117764742051, −8.295463230232468680708267546667, −8.286577858241470359999842963459, −7.44634313053002225289393409560, −7.14258485507133526607743722122, −6.75382417627942756480321655215, −6.38594470424405032777692483958, −5.65321199041131712001001627737, −5.27951855114179207636657680047, −4.51627240247470658040113759297, −4.42485224867946784061644121196, −3.66436634018199143084147189566, −3.60165692141157611830664999333, −2.65069316426082859006632021710, −1.99670220402920670312030520364, −1.87763258882109013351049874304, −0.64981183890686412160020465407,
0.64981183890686412160020465407, 1.87763258882109013351049874304, 1.99670220402920670312030520364, 2.65069316426082859006632021710, 3.60165692141157611830664999333, 3.66436634018199143084147189566, 4.42485224867946784061644121196, 4.51627240247470658040113759297, 5.27951855114179207636657680047, 5.65321199041131712001001627737, 6.38594470424405032777692483958, 6.75382417627942756480321655215, 7.14258485507133526607743722122, 7.44634313053002225289393409560, 8.286577858241470359999842963459, 8.295463230232468680708267546667, 9.026045909869793674117764742051, 9.117789009467200424218151363662, 9.324299252313651527737534493362, 9.419090959450620177767908620279
