| L(s) = 1 | + 2·3-s − 4-s + 4·7-s + 9-s − 2·12-s + 3·13-s − 3·16-s − 6·19-s + 8·21-s − 10·25-s − 4·27-s − 4·28-s + 14·31-s − 36-s + 12·37-s + 6·39-s − 2·43-s − 6·48-s + 9·49-s − 3·52-s − 12·57-s + 20·61-s + 4·63-s + 7·64-s + 2·67-s + 8·73-s − 20·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 0.832·13-s − 3/4·16-s − 1.37·19-s + 1.74·21-s − 2·25-s − 0.769·27-s − 0.755·28-s + 2.51·31-s − 1/6·36-s + 1.97·37-s + 0.960·39-s − 0.304·43-s − 0.866·48-s + 9/7·49-s − 0.416·52-s − 1.58·57-s + 2.56·61-s + 0.503·63-s + 7/8·64-s + 0.244·67-s + 0.936·73-s − 2.30·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.147971377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.147971377\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.150275188299931969665499139005, −8.049280863442422348555811087408, −7.59450943185977558854350542793, −6.91372503677898216326468883692, −6.32518496511319018120119356276, −6.05721856581736372807458527004, −5.39333818781539371489322753563, −4.83464076593295898441308567277, −4.32455597829515584173555719046, −4.07888231525233385585312014709, −3.60838848875325386242678924927, −2.64276618569156944120174936178, −2.29724410428618923837380073091, −1.78124924777939950511017610541, −0.800105156383902302859507969562,
0.800105156383902302859507969562, 1.78124924777939950511017610541, 2.29724410428618923837380073091, 2.64276618569156944120174936178, 3.60838848875325386242678924927, 4.07888231525233385585312014709, 4.32455597829515584173555719046, 4.83464076593295898441308567277, 5.39333818781539371489322753563, 6.05721856581736372807458527004, 6.32518496511319018120119356276, 6.91372503677898216326468883692, 7.59450943185977558854350542793, 8.049280863442422348555811087408, 8.150275188299931969665499139005
