| L(s) = 1 | − 2·2-s + 3·4-s + 5-s − 4·8-s − 2·10-s + 11-s − 13-s + 5·16-s + 3·20-s − 2·22-s + 23-s + 2·26-s + 29-s − 31-s − 6·32-s + 2·37-s − 4·40-s + 41-s + 3·44-s − 2·46-s + 2·49-s − 3·52-s + 55-s − 2·58-s − 61-s + 2·62-s + 7·64-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 5-s − 4·8-s − 2·10-s + 11-s − 13-s + 5·16-s + 3·20-s − 2·22-s + 23-s + 2·26-s + 29-s − 31-s − 6·32-s + 2·37-s − 4·40-s + 41-s + 3·44-s − 2·46-s + 2·49-s − 3·52-s + 55-s − 2·58-s − 61-s + 2·62-s + 7·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7124224564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7124224564\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.136708324390260084295474140717, −8.979374712579495712556158665599, −8.642120723549227288324274161930, −8.274634844347233882051705482470, −7.50120339307721629264083526537, −7.48468361887103358658187060650, −7.16418426001485927859536252927, −6.74829462531230026047386961678, −6.16502955346187186860172135688, −5.99462427474064995080036242492, −5.68506838850060842044246907037, −5.17038634608667713165506486556, −4.32457938562147200984287424066, −4.17279483487520558783943637566, −3.08923191260175614729237940586, −2.94296595494778320223885697617, −2.43173642564324971593775020106, −1.89936002924694953236996173361, −1.38723021497164378333183857301, −0.811964214674075218398817175069,
0.811964214674075218398817175069, 1.38723021497164378333183857301, 1.89936002924694953236996173361, 2.43173642564324971593775020106, 2.94296595494778320223885697617, 3.08923191260175614729237940586, 4.17279483487520558783943637566, 4.32457938562147200984287424066, 5.17038634608667713165506486556, 5.68506838850060842044246907037, 5.99462427474064995080036242492, 6.16502955346187186860172135688, 6.74829462531230026047386961678, 7.16418426001485927859536252927, 7.48468361887103358658187060650, 7.50120339307721629264083526537, 8.274634844347233882051705482470, 8.642120723549227288324274161930, 8.979374712579495712556158665599, 9.136708324390260084295474140717
