| L(s) = 1 | + 2-s + 4-s + 3·5-s + 2·7-s + 8-s − 9-s + 3·10-s − 2·13-s + 2·14-s + 16-s − 18-s + 3·20-s + 4·25-s − 2·26-s + 2·28-s + 32-s + 6·35-s − 36-s + 2·37-s + 3·40-s − 3·45-s − 10·47-s − 7·49-s + 4·50-s − 2·52-s + 2·56-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.755·7-s + 0.353·8-s − 1/3·9-s + 0.948·10-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.235·18-s + 0.670·20-s + 4/5·25-s − 0.392·26-s + 0.377·28-s + 0.176·32-s + 1.01·35-s − 1/6·36-s + 0.328·37-s + 0.474·40-s − 0.447·45-s − 1.45·47-s − 49-s + 0.565·50-s − 0.277·52-s + 0.267·56-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.890704017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.890704017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.925181025005833152475597219540, −9.490422435360190412247163526048, −8.900832316813910546464957105762, −8.281589094984078812692544079379, −7.86262323703768997396877892390, −7.16722420622100988834428197118, −6.61596321721506488375751420916, −6.09429344794924968828988933771, −5.59200537471275222346671912296, −5.01180052215032667025310076262, −4.68200613623420218940878034009, −3.76048468109763736328041177461, −2.91112609366534338189655955750, −2.22860523628666793533249343021, −1.51291789026756271512295453365,
1.51291789026756271512295453365, 2.22860523628666793533249343021, 2.91112609366534338189655955750, 3.76048468109763736328041177461, 4.68200613623420218940878034009, 5.01180052215032667025310076262, 5.59200537471275222346671912296, 6.09429344794924968828988933771, 6.61596321721506488375751420916, 7.16722420622100988834428197118, 7.86262323703768997396877892390, 8.281589094984078812692544079379, 8.900832316813910546464957105762, 9.490422435360190412247163526048, 9.925181025005833152475597219540
