| L(s) = 1 | + 3-s + 4·5-s + 9-s + 4·15-s − 8·19-s + 2·25-s + 27-s + 4·29-s + 8·43-s + 4·45-s + 16·47-s + 2·49-s + 20·53-s − 8·57-s + 8·67-s − 32·71-s − 12·73-s + 2·75-s + 81-s + 4·87-s − 32·95-s − 28·97-s − 12·101-s − 6·121-s − 28·125-s + 127-s + 8·129-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.83·19-s + 2/5·25-s + 0.192·27-s + 0.742·29-s + 1.21·43-s + 0.596·45-s + 2.33·47-s + 2/7·49-s + 2.74·53-s − 1.05·57-s + 0.977·67-s − 3.79·71-s − 1.40·73-s + 0.230·75-s + 1/9·81-s + 0.428·87-s − 3.28·95-s − 2.84·97-s − 1.19·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240834063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240834063\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.08615312505774861350749378685, −9.488699237661810053920246470193, −8.844581178668094717437191348011, −8.839886292016704738441339481672, −8.093022628739907221893201157530, −7.33135946825865482348621844071, −6.91469103626909406223255043804, −6.15267013015603292112906972931, −5.82131382158921014822362291005, −5.37335304530331906156568812203, −4.20875443528614711634884665822, −4.10625798014363516984369255368, −2.59447091927996366543258857368, −2.44821729992400328434125228202, −1.47354540780337201919328149138,
1.47354540780337201919328149138, 2.44821729992400328434125228202, 2.59447091927996366543258857368, 4.10625798014363516984369255368, 4.20875443528614711634884665822, 5.37335304530331906156568812203, 5.82131382158921014822362291005, 6.15267013015603292112906972931, 6.91469103626909406223255043804, 7.33135946825865482348621844071, 8.093022628739907221893201157530, 8.839886292016704738441339481672, 8.844581178668094717437191348011, 9.488699237661810053920246470193, 10.08615312505774861350749378685
