| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s + 7-s − 8-s + 9-s + 4·10-s + 12-s − 5·13-s − 14-s − 4·15-s + 16-s − 3·17-s − 18-s + 3·19-s − 4·20-s + 21-s + 23-s − 24-s + 6·25-s + 5·26-s + 4·27-s + 28-s + 3·29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s − 1.38·13-s − 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.688·19-s − 0.894·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 6/5·25-s + 0.980·26-s + 0.769·27-s + 0.188·28-s + 0.557·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4064521914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4064521914\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 72 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.6470521258, −19.1983346540, −18.8138416831, −18.0339280685, −17.5824210060, −16.8343970233, −16.2940991954, −15.5725412197, −15.3743062854, −14.7834792262, −14.1505771079, −13.3619212628, −12.4215030100, −11.8651183180, −11.6362096375, −10.6602566748, −10.0971312048, −9.13832703823, −8.56559477720, −7.76746543474, −7.47625192182, −6.65624212333, −4.99382097136, −4.10508156552, −2.81493669109,
2.81493669109, 4.10508156552, 4.99382097136, 6.65624212333, 7.47625192182, 7.76746543474, 8.56559477720, 9.13832703823, 10.0971312048, 10.6602566748, 11.6362096375, 11.8651183180, 12.4215030100, 13.3619212628, 14.1505771079, 14.7834792262, 15.3743062854, 15.5725412197, 16.2940991954, 16.8343970233, 17.5824210060, 18.0339280685, 18.8138416831, 19.1983346540, 19.6470521258
