| L(s) = 1 | + 3·3-s + 6·9-s + 6·13-s − 4·16-s + 12·19-s + 25-s + 9·27-s + 12·31-s + 18·39-s − 12·43-s − 12·48-s + 36·57-s − 28·67-s + 12·73-s + 3·75-s − 2·79-s + 9·81-s + 36·93-s − 30·97-s − 18·103-s − 30·109-s + 36·117-s − 21·121-s + 127-s − 36·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 1.66·13-s − 16-s + 2.75·19-s + 1/5·25-s + 1.73·27-s + 2.15·31-s + 2.88·39-s − 1.82·43-s − 1.73·48-s + 4.76·57-s − 3.42·67-s + 1.40·73-s + 0.346·75-s − 0.225·79-s + 81-s + 3.73·93-s − 3.04·97-s − 1.77·103-s − 2.87·109-s + 3.32·117-s − 1.90·121-s + 0.0887·127-s − 3.16·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.555608806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.555608806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
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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
−8.252623699886366464992405726957, −8.210184381994137443519230420468, −7.79509008882088804636414623325, −7.18674908490146366638229016691, −6.67134380816033742861497166759, −6.50048591700478498626320913392, −5.54795317274704426653960185894, −5.25756591986209044151466901261, −4.33023383010768085745155570778, −4.20509313138730818892241036590, −3.31721807269730857488302455154, −3.07565995342284756040604408072, −2.65178092612008457940193320214, −1.56775892122303114558742737725, −1.20406663212713068028112418239,
1.20406663212713068028112418239, 1.56775892122303114558742737725, 2.65178092612008457940193320214, 3.07565995342284756040604408072, 3.31721807269730857488302455154, 4.20509313138730818892241036590, 4.33023383010768085745155570778, 5.25756591986209044151466901261, 5.54795317274704426653960185894, 6.50048591700478498626320913392, 6.67134380816033742861497166759, 7.18674908490146366638229016691, 7.79509008882088804636414623325, 8.210184381994137443519230420468, 8.252623699886366464992405726957
