| L(s) = 1 | − 6·7-s − 3·9-s − 9·13-s + 8·19-s − 10·25-s + 3·31-s − 10·37-s + 17·49-s − 27·61-s + 18·63-s − 11·67-s − 7·73-s − 21·79-s + 9·81-s + 54·91-s − 33·97-s + 20·103-s + 27·117-s + 11·121-s + 127-s + 131-s − 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 9-s − 2.49·13-s + 1.83·19-s − 2·25-s + 0.538·31-s − 1.64·37-s + 17/7·49-s − 3.45·61-s + 2.26·63-s − 1.34·67-s − 0.819·73-s − 2.36·79-s + 81-s + 5.66·91-s − 3.35·97-s + 1.97·103-s + 2.49·117-s + 121-s + 0.0887·127-s + 0.0873·131-s − 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 67 | $C_2$ | \( 1 + 11 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.753262822512613540176371550607, −9.669008988730103615969072931109, −9.520509593862461426931722994725, −8.889363340673095284466500425189, −8.473594594990375988546197641721, −7.60658930531610968604918821978, −7.38647807939077263542889296527, −7.28615047395844851150059915793, −6.31460259250333074662458331719, −6.30624219427902568529593998733, −5.50974263582074932377974302382, −5.38562577314389272330452197398, −4.64328377949172704156158786751, −4.03518216850404759940132545039, −3.24683755246149025095292081906, −2.98916828815000323641775569873, −2.67597552725336831600453735092, −1.69955755213743297015120101622, 0, 0,
1.69955755213743297015120101622, 2.67597552725336831600453735092, 2.98916828815000323641775569873, 3.24683755246149025095292081906, 4.03518216850404759940132545039, 4.64328377949172704156158786751, 5.38562577314389272330452197398, 5.50974263582074932377974302382, 6.30624219427902568529593998733, 6.31460259250333074662458331719, 7.28615047395844851150059915793, 7.38647807939077263542889296527, 7.60658930531610968604918821978, 8.473594594990375988546197641721, 8.889363340673095284466500425189, 9.520509593862461426931722994725, 9.669008988730103615969072931109, 9.753262822512613540176371550607
