| L(s) = 1 | − 4·7-s + 9-s + 12·17-s − 8·23-s − 10·25-s − 20·31-s − 4·41-s + 24·47-s − 2·49-s − 4·63-s + 8·71-s − 20·73-s + 12·79-s + 81-s + 4·89-s − 12·97-s + 20·103-s − 28·113-s − 48·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1/3·9-s + 2.91·17-s − 1.66·23-s − 2·25-s − 3.59·31-s − 0.624·41-s + 3.50·47-s − 2/7·49-s − 0.503·63-s + 0.949·71-s − 2.34·73-s + 1.35·79-s + 1/9·81-s + 0.423·89-s − 1.21·97-s + 1.97·103-s − 2.63·113-s − 4.40·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−9.355375973689939441990781827382, −8.658903060264415544055202217864, −7.73185283482156320096864018539, −7.70036027517747780731408099423, −7.29744951786402483785793531451, −6.52243750033071865852983380426, −5.81271121311664380914911812156, −5.75568326693151721019291253980, −5.25073137830748218691985779847, −3.94434944725762226728078180666, −3.77490371552045331982722842815, −3.36330701716196318281397292070, −2.36547685273488160999702672858, −1.50413098873776121963751941726, 0,
1.50413098873776121963751941726, 2.36547685273488160999702672858, 3.36330701716196318281397292070, 3.77490371552045331982722842815, 3.94434944725762226728078180666, 5.25073137830748218691985779847, 5.75568326693151721019291253980, 5.81271121311664380914911812156, 6.52243750033071865852983380426, 7.29744951786402483785793531451, 7.70036027517747780731408099423, 7.73185283482156320096864018539, 8.658903060264415544055202217864, 9.355375973689939441990781827382
