| L(s) = 1 | − 3·5-s − 7·19-s + 3·23-s + 4·25-s + 15·29-s − 15·43-s − 2·49-s − 3·53-s − 3·67-s − 3·71-s + 18·73-s + 21·95-s + 12·97-s − 12·101-s − 9·115-s + 121-s + 3·125-s + 127-s + 131-s + 137-s + 139-s − 45·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.60·19-s + 0.625·23-s + 4/5·25-s + 2.78·29-s − 2.28·43-s − 2/7·49-s − 0.412·53-s − 0.366·67-s − 0.356·71-s + 2.10·73-s + 2.15·95-s + 1.21·97-s − 1.19·101-s − 0.839·115-s + 1/11·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.73·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 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
−8.390458744732562282801165925174, −7.86321354187183783039895204895, −7.48014655646988318375931244152, −6.74962989482314584805541021995, −6.55377070108065675411040352814, −6.23414241184305987199807199456, −5.24802402083850128266850887580, −4.86777783855550131563533862529, −4.45259921660819926503150074110, −3.94686773224235125859275206009, −3.34112097926251195216790938275, −2.85773074773182498526910783887, −2.09088263047254841529058600267, −1.07386153798253852754348245763, 0,
1.07386153798253852754348245763, 2.09088263047254841529058600267, 2.85773074773182498526910783887, 3.34112097926251195216790938275, 3.94686773224235125859275206009, 4.45259921660819926503150074110, 4.86777783855550131563533862529, 5.24802402083850128266850887580, 6.23414241184305987199807199456, 6.55377070108065675411040352814, 6.74962989482314584805541021995, 7.48014655646988318375931244152, 7.86321354187183783039895204895, 8.390458744732562282801165925174
