| L(s) = 1 | − 2·3-s + 9-s − 12·11-s + 4·13-s + 12·23-s − 6·25-s + 4·27-s + 24·33-s + 12·37-s − 8·39-s − 10·49-s − 16·59-s + 28·61-s − 24·69-s + 4·71-s − 28·73-s + 12·75-s − 11·81-s + 16·83-s + 4·97-s − 12·99-s − 36·107-s + 12·109-s − 24·111-s + 4·117-s + 86·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 3.61·11-s + 1.10·13-s + 2.50·23-s − 6/5·25-s + 0.769·27-s + 4.17·33-s + 1.97·37-s − 1.28·39-s − 1.42·49-s − 2.08·59-s + 3.58·61-s − 2.88·69-s + 0.474·71-s − 3.27·73-s + 1.38·75-s − 1.22·81-s + 1.75·83-s + 0.406·97-s − 1.20·99-s − 3.48·107-s + 1.14·109-s − 2.27·111-s + 0.369·117-s + 7.81·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.306702051055435143424050976414, −8.143686641212330787302129145795, −7.56035654213918809032647626203, −7.22959647173292270161762928560, −6.55244625527635278375850497006, −5.94766297465593937866721260525, −5.68695398299974094866219608459, −5.12427258054127542705663030395, −4.93933220983461313213877514067, −4.31537121465042302044065022649, −3.24743736182506202340620349352, −2.88162263616814243903499687246, −2.26722510774655753380953940094, −0.989974284560430244131776641416, 0,
0.989974284560430244131776641416, 2.26722510774655753380953940094, 2.88162263616814243903499687246, 3.24743736182506202340620349352, 4.31537121465042302044065022649, 4.93933220983461313213877514067, 5.12427258054127542705663030395, 5.68695398299974094866219608459, 5.94766297465593937866721260525, 6.55244625527635278375850497006, 7.22959647173292270161762928560, 7.56035654213918809032647626203, 8.143686641212330787302129145795, 8.306702051055435143424050976414
