| L(s) = 1 | − 2·4-s − 4·7-s + 9-s + 2·11-s − 2·13-s + 4·16-s + 4·17-s − 4·19-s + 8·23-s − 5·25-s + 8·28-s − 6·29-s + 2·31-s − 2·36-s − 10·37-s − 4·41-s − 12·43-s − 4·44-s + 8·47-s + 2·49-s + 4·52-s − 2·59-s − 14·61-s − 4·63-s − 8·64-s − 4·67-s − 8·68-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 16-s + 0.970·17-s − 0.917·19-s + 1.66·23-s − 25-s + 1.51·28-s − 1.11·29-s + 0.359·31-s − 1/3·36-s − 1.64·37-s − 0.624·41-s − 1.82·43-s − 0.603·44-s + 1.16·47-s + 2/7·49-s + 0.554·52-s − 0.260·59-s − 1.79·61-s − 0.503·63-s − 64-s − 0.488·67-s − 0.970·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.8595826501, −14.3820983435, −13.7410216765, −13.4512544705, −13.1203440067, −12.5766943432, −12.2093864102, −11.9594843409, −11.0713887673, −10.5604845975, −10.0406635436, −9.69164586099, −9.34365210340, −8.81028299029, −8.40099932774, −7.54559136120, −7.21584134111, −6.48002006446, −6.13128908222, −5.26829567139, −4.91780353436, −3.97563996350, −3.54080710729, −2.96031601073, −1.57454212811, 0,
1.57454212811, 2.96031601073, 3.54080710729, 3.97563996350, 4.91780353436, 5.26829567139, 6.13128908222, 6.48002006446, 7.21584134111, 7.54559136120, 8.40099932774, 8.81028299029, 9.34365210340, 9.69164586099, 10.0406635436, 10.5604845975, 11.0713887673, 11.9594843409, 12.2093864102, 12.5766943432, 13.1203440067, 13.4512544705, 13.7410216765, 14.3820983435, 14.8595826501
