| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 7-s − 4·8-s + 3·9-s + 7·11-s − 6·12-s + 4·13-s + 2·14-s + 5·16-s − 2·17-s − 6·18-s − 7·19-s + 2·21-s − 14·22-s − 23-s + 8·24-s − 8·26-s − 4·27-s − 3·28-s − 6·29-s − 2·31-s − 6·32-s − 14·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s + 2.11·11-s − 1.73·12-s + 1.10·13-s + 0.534·14-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 1.60·19-s + 0.436·21-s − 2.98·22-s − 0.208·23-s + 1.63·24-s − 1.56·26-s − 0.769·27-s − 0.566·28-s − 1.11·29-s − 0.359·31-s − 1.06·32-s − 2.43·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 90 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 134 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 206 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.187739174110367421553311030345, −7.84418435609534387069938295038, −7.28076407563719246707254811944, −6.94043384359763290796180965102, −6.68804144562419707163809570057, −6.44554811274053435676553966938, −6.05975329508170626646811782590, −5.96301252568829614000538049627, −5.28458938930947918254930968881, −4.86624348356107088704035931927, −4.32630470800383424636289952301, −3.91762770278380212533946480219, −3.42504671715502040127430119384, −3.39198590475853754033954296876, −2.26566659218720602915842411752, −1.85955530219704585871673093711, −1.48270525370762976810609932868, −1.10700420719223053737421382066, 0, 0,
1.10700420719223053737421382066, 1.48270525370762976810609932868, 1.85955530219704585871673093711, 2.26566659218720602915842411752, 3.39198590475853754033954296876, 3.42504671715502040127430119384, 3.91762770278380212533946480219, 4.32630470800383424636289952301, 4.86624348356107088704035931927, 5.28458938930947918254930968881, 5.96301252568829614000538049627, 6.05975329508170626646811782590, 6.44554811274053435676553966938, 6.68804144562419707163809570057, 6.94043384359763290796180965102, 7.28076407563719246707254811944, 7.84418435609534387069938295038, 8.187739174110367421553311030345
