| L(s) = 1 | − 6·9-s − 12·11-s − 4·19-s − 16·31-s − 24·41-s − 12·49-s − 24·59-s − 20·61-s − 24·71-s + 8·79-s + 14·81-s − 24·89-s + 72·99-s + 12·101-s + 8·109-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 24·171-s + ⋯ |
| L(s) = 1 | − 2·9-s − 3.61·11-s − 0.917·19-s − 2.87·31-s − 3.74·41-s − 1.71·49-s − 3.12·59-s − 2.56·61-s − 2.84·71-s + 0.900·79-s + 14/9·81-s − 2.54·89-s + 7.23·99-s + 1.19·101-s + 0.766·109-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T^{2} + 22 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 6 T^{2} + 302 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 106 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 662 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 94 T^{2} + 4542 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 158 T^{2} + 11454 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 142 T^{2} + 10374 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 5622 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 116 T^{2} + 10662 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_4$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 174 T^{2} + 18782 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.16939286754402311942695538044, −6.75373938335604072244306500622, −6.52365139204136561985983629483, −6.10290985005186838732174441307, −6.07002590238187895766504886816, −5.91407589479844595962645732809, −5.73881253458268546942884468386, −5.44673960962651922960646582450, −5.40647311620450029780470179093, −4.89902955053939527150703210761, −4.84282607680130750127712398673, −4.78452359403743836035818973320, −4.74208212326978026722927022325, −4.16919777626821001697467686248, −3.84981833633243038529077997270, −3.46030256124295743509377535450, −3.26110465758025879119597275130, −3.18138131996491837060870976958, −3.03408814518767089601946980012, −2.66207570211518779574709484392, −2.51339429226187543878896182769, −2.03770108810668503165609429346, −2.01505303799945403227893738935, −1.45148435900293104470555652400, −1.42039768211706731557935066978, 0, 0, 0, 0,
1.42039768211706731557935066978, 1.45148435900293104470555652400, 2.01505303799945403227893738935, 2.03770108810668503165609429346, 2.51339429226187543878896182769, 2.66207570211518779574709484392, 3.03408814518767089601946980012, 3.18138131996491837060870976958, 3.26110465758025879119597275130, 3.46030256124295743509377535450, 3.84981833633243038529077997270, 4.16919777626821001697467686248, 4.74208212326978026722927022325, 4.78452359403743836035818973320, 4.84282607680130750127712398673, 4.89902955053939527150703210761, 5.40647311620450029780470179093, 5.44673960962651922960646582450, 5.73881253458268546942884468386, 5.91407589479844595962645732809, 6.07002590238187895766504886816, 6.10290985005186838732174441307, 6.52365139204136561985983629483, 6.75373938335604072244306500622, 7.16939286754402311942695538044
