L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 2·14-s + 16-s + 12·23-s − 25-s − 2·28-s + 10·31-s + 32-s + 12·41-s + 12·46-s − 12·47-s − 11·49-s − 50-s − 2·56-s + 10·62-s + 64-s − 14·73-s + 16·79-s + 12·82-s + 36·89-s + 12·92-s − 12·94-s − 2·97-s − 11·98-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.534·14-s + 1/4·16-s + 2.50·23-s − 1/5·25-s − 0.377·28-s + 1.79·31-s + 0.176·32-s + 1.87·41-s + 1.76·46-s − 1.75·47-s − 1.57·49-s − 0.141·50-s − 0.267·56-s + 1.27·62-s + 1/8·64-s − 1.63·73-s + 1.80·79-s + 1.32·82-s + 3.81·89-s + 1.25·92-s − 1.23·94-s − 0.203·97-s − 1.11·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.252789771\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.252789771\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.512741326522107701681952133776, −9.254274842104821367212382551306, −8.728358107276488427256715933065, −7.82723843094092867969941089213, −7.78966227402229455461895642068, −6.71508725988629390136097161445, −6.69450931342145073952637354839, −6.11240325026052414300507082707, −5.39588744112665066773671436635, −4.77709333112371596512170551343, −4.45861885513123914392518249053, −3.38549178369413487864105189864, −3.13246179353932073023232514138, −2.34838622997525881980959331572, −1.07855163027675466308171882838,
1.07855163027675466308171882838, 2.34838622997525881980959331572, 3.13246179353932073023232514138, 3.38549178369413487864105189864, 4.45861885513123914392518249053, 4.77709333112371596512170551343, 5.39588744112665066773671436635, 6.11240325026052414300507082707, 6.69450931342145073952637354839, 6.71508725988629390136097161445, 7.78966227402229455461895642068, 7.82723843094092867969941089213, 8.728358107276488427256715933065, 9.254274842104821367212382551306, 9.512741326522107701681952133776