| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s + 4·7-s − 4·8-s − 4·9-s − 11-s + 3·12-s − 6·13-s − 8·14-s + 5·16-s − 6·17-s + 8·18-s − 2·19-s + 4·21-s + 2·22-s + 6·23-s − 4·24-s − 5·25-s + 12·26-s − 6·27-s + 12·28-s − 7·29-s + 3·31-s − 6·32-s − 33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.51·7-s − 1.41·8-s − 4/3·9-s − 0.301·11-s + 0.866·12-s − 1.66·13-s − 2.13·14-s + 5/4·16-s − 1.45·17-s + 1.88·18-s − 0.458·19-s + 0.872·21-s + 0.426·22-s + 1.25·23-s − 0.816·24-s − 25-s + 2.35·26-s − 1.15·27-s + 2.26·28-s − 1.29·29-s + 0.538·31-s − 1.06·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64288324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64288324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| 211 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 59 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 217 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 133 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 157 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 193 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.74305080200683589186518069485, −7.55201090005059828870232232128, −7.12817535973137156964291299412, −6.87774095289158745493748963022, −6.33089491160257623395421785306, −6.09729070909545258929629001577, −5.48184227509237149910406286476, −5.32356618169937814556118154033, −4.85015858437178472902272334762, −4.62956624010254801722489925914, −4.03598559035604298697352667156, −3.63313256278593732921444578019, −2.86861698525612983896027321234, −2.80276284129701785869264019402, −2.31920242341318240746255553032, −1.92773251985947286974626800766, −1.73392881987348740330249936915, −0.910625815505891104699746177698, 0, 0,
0.910625815505891104699746177698, 1.73392881987348740330249936915, 1.92773251985947286974626800766, 2.31920242341318240746255553032, 2.80276284129701785869264019402, 2.86861698525612983896027321234, 3.63313256278593732921444578019, 4.03598559035604298697352667156, 4.62956624010254801722489925914, 4.85015858437178472902272334762, 5.32356618169937814556118154033, 5.48184227509237149910406286476, 6.09729070909545258929629001577, 6.33089491160257623395421785306, 6.87774095289158745493748963022, 7.12817535973137156964291299412, 7.55201090005059828870232232128, 7.74305080200683589186518069485
