| L(s) = 1 | + 3-s − 2·4-s − 2·7-s − 2·9-s − 2·12-s + 13-s − 7·19-s − 2·21-s − 2·25-s − 5·27-s + 4·28-s − 2·31-s + 4·36-s + 15·37-s + 39-s + 21·43-s + 3·49-s − 2·52-s − 7·57-s + 61-s + 4·63-s + 8·64-s + 5·67-s + 16·73-s − 2·75-s + 14·76-s − 27·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.755·7-s − 2/3·9-s − 0.577·12-s + 0.277·13-s − 1.60·19-s − 0.436·21-s − 2/5·25-s − 0.962·27-s + 0.755·28-s − 0.359·31-s + 2/3·36-s + 2.46·37-s + 0.160·39-s + 3.20·43-s + 3/7·49-s − 0.277·52-s − 0.927·57-s + 0.128·61-s + 0.503·63-s + 64-s + 0.610·67-s + 1.87·73-s − 0.230·75-s + 1.60·76-s − 3.03·79-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.175103398471717001174795995142, −7.60152577513698248037384881289, −7.20093367475201307681764379924, −6.48891849967434372347939513893, −6.18251082148931620042689407809, −5.71166117532807465223075397170, −5.36825183723845399742565356566, −4.39181834955626270018660051830, −4.28156071781390340579393795057, −3.87131895817643317945492144748, −3.16264691496313075075906095461, −2.49372941601990847213775271161, −2.24368424699872810368910669445, −0.926622631406678675659944923079, 0,
0.926622631406678675659944923079, 2.24368424699872810368910669445, 2.49372941601990847213775271161, 3.16264691496313075075906095461, 3.87131895817643317945492144748, 4.28156071781390340579393795057, 4.39181834955626270018660051830, 5.36825183723845399742565356566, 5.71166117532807465223075397170, 6.18251082148931620042689407809, 6.48891849967434372347939513893, 7.20093367475201307681764379924, 7.60152577513698248037384881289, 8.175103398471717001174795995142
