L(s) = 1 | + 2·2-s + 3·4-s + 3·7-s + 4·8-s − 5·9-s + 4·11-s + 6·14-s + 5·16-s − 10·18-s + 8·22-s − 2·23-s + 6·25-s + 9·28-s − 10·29-s + 6·32-s − 15·36-s − 4·37-s + 8·43-s + 12·44-s − 4·46-s + 2·49-s + 12·50-s − 2·53-s + 12·56-s − 20·58-s − 15·63-s + 7·64-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 5/3·9-s + 1.20·11-s + 1.60·14-s + 5/4·16-s − 2.35·18-s + 1.70·22-s − 0.417·23-s + 6/5·25-s + 1.70·28-s − 1.85·29-s + 1.06·32-s − 5/2·36-s − 0.657·37-s + 1.21·43-s + 1.80·44-s − 0.589·46-s + 2/7·49-s + 1.69·50-s − 0.274·53-s + 1.60·56-s − 2.62·58-s − 1.88·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.647568683\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.647568683\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.856310223818109751782129963913, −9.254038079040445697892143439151, −8.655339283587140731226502868667, −8.408486909011940238981299566497, −7.60323894240594246780554900498, −7.25038064772684706053554512263, −6.45399284641636243911038304495, −6.09315132100286987427486982952, −5.29912825884459151712444657250, −5.28069074518443602737935578596, −4.32145140400317761773094051986, −3.86057015395193111701613139713, −3.13458058385460483521784962114, −2.39336202780458240475893030948, −1.50913345329536755396223442365,
1.50913345329536755396223442365, 2.39336202780458240475893030948, 3.13458058385460483521784962114, 3.86057015395193111701613139713, 4.32145140400317761773094051986, 5.28069074518443602737935578596, 5.29912825884459151712444657250, 6.09315132100286987427486982952, 6.45399284641636243911038304495, 7.25038064772684706053554512263, 7.60323894240594246780554900498, 8.408486909011940238981299566497, 8.655339283587140731226502868667, 9.254038079040445697892143439151, 9.856310223818109751782129963913