| L(s) = 1 | + 2·3-s + 3·9-s + 8·11-s + 12·17-s − 10·25-s + 4·27-s + 16·33-s − 4·41-s − 16·43-s − 10·49-s + 24·51-s + 8·59-s − 8·67-s − 20·73-s − 20·75-s + 5·81-s − 24·83-s + 4·89-s − 12·97-s + 24·99-s + 24·107-s − 28·113-s + 26·121-s − 8·123-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 2.41·11-s + 2.91·17-s − 2·25-s + 0.769·27-s + 2.78·33-s − 0.624·41-s − 2.43·43-s − 1.42·49-s + 3.36·51-s + 1.04·59-s − 0.977·67-s − 2.34·73-s − 2.30·75-s + 5/9·81-s − 2.63·83-s + 0.423·89-s − 1.21·97-s + 2.41·99-s + 2.32·107-s − 2.63·113-s + 2.36·121-s − 0.721·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.035318132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.035318132\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.464663439718228165986962411065, −8.658903060264415544055202217864, −8.503024042291174545390595385787, −7.70036027517747780731408099423, −7.67554775307177371631314255080, −6.80910210319960057293306406499, −6.52243750033071865852983380426, −5.75568326693151721019291253980, −5.36858848597467135489376840463, −4.36074490911347519116057298154, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −3.06807634409715093203408940942, −1.59108885184574423909298130831, −1.50413098873776121963751941726,
1.50413098873776121963751941726, 1.59108885184574423909298130831, 3.06807634409715093203408940942, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 4.36074490911347519116057298154, 5.36858848597467135489376840463, 5.75568326693151721019291253980, 6.52243750033071865852983380426, 6.80910210319960057293306406499, 7.67554775307177371631314255080, 7.70036027517747780731408099423, 8.503024042291174545390595385787, 8.658903060264415544055202217864, 9.464663439718228165986962411065
