| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s − 9-s − 2·10-s + 5·11-s − 12-s − 6·13-s + 2·14-s + 2·15-s + 16-s − 3·17-s − 18-s − 3·19-s − 2·20-s − 2·21-s + 5·22-s − 2·23-s − 24-s − 25-s − 6·26-s + 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 1/3·9-s − 0.632·10-s + 1.50·11-s − 0.288·12-s − 1.66·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.688·19-s − 0.447·20-s − 0.436·21-s + 1.06·22-s − 0.417·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8447804386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8447804386\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−17.8386888090, −17.5442935528, −17.0406118416, −16.6830275380, −16.0375643724, −15.3376859404, −14.8992914148, −14.4951797316, −14.0916295383, −13.2859500860, −12.4635176254, −12.1027752387, −11.6716606275, −11.2189813326, −10.6912421483, −9.76484458202, −9.02826212357, −8.31851072904, −7.39410638864, −7.07901565704, −6.06748159208, −5.38854611424, −4.30327828737, −4.06396508747, −2.35093004831,
2.35093004831, 4.06396508747, 4.30327828737, 5.38854611424, 6.06748159208, 7.07901565704, 7.39410638864, 8.31851072904, 9.02826212357, 9.76484458202, 10.6912421483, 11.2189813326, 11.6716606275, 12.1027752387, 12.4635176254, 13.2859500860, 14.0916295383, 14.4951797316, 14.8992914148, 15.3376859404, 16.0375643724, 16.6830275380, 17.0406118416, 17.5442935528, 17.8386888090
