| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 2·5-s + 2·6-s − 3·7-s + 4·10-s − 6·11-s − 2·12-s + 6·14-s + 2·15-s − 4·16-s + 19-s − 4·20-s + 3·21-s + 12·22-s − 4·23-s − 25-s + 27-s − 6·28-s − 4·30-s − 31-s + 8·32-s + 6·33-s + 6·35-s + 3·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.26·10-s − 1.80·11-s − 0.577·12-s + 1.60·14-s + 0.516·15-s − 16-s + 0.229·19-s − 0.894·20-s + 0.654·21-s + 2.55·22-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.13·28-s − 0.730·30-s − 0.179·31-s + 1.41·32-s + 1.04·33-s + 1.01·35-s + 0.493·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T - 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 109 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 63 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.3558532400, −17.7556587135, −17.4000248614, −16.5497040857, −16.3003492118, −15.8822642659, −15.5694481617, −14.9005046449, −13.9322788477, −13.2224317670, −12.9591529596, −12.0769467458, −11.6723913485, −10.9065175172, −10.4935156562, −9.91760295678, −9.46769940710, −8.51561782979, −8.02635605633, −7.48404734277, −6.81729676334, −5.92734365630, −5.05963353645, −3.90547747157, −2.63341570517, 0,
2.63341570517, 3.90547747157, 5.05963353645, 5.92734365630, 6.81729676334, 7.48404734277, 8.02635605633, 8.51561782979, 9.46769940710, 9.91760295678, 10.4935156562, 10.9065175172, 11.6723913485, 12.0769467458, 12.9591529596, 13.2224317670, 13.9322788477, 14.9005046449, 15.5694481617, 15.8822642659, 16.3003492118, 16.5497040857, 17.4000248614, 17.7556587135, 18.3558532400
