| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 2·9-s − 10-s + 11-s − 12-s − 3·13-s + 14-s + 15-s + 16-s − 8·17-s + 2·18-s + 19-s − 20-s − 21-s + 22-s + 7·23-s − 24-s − 6·25-s − 3·26-s − 6·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.45·23-s − 0.204·24-s − 6/5·25-s − 0.588·26-s − 1.15·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9820145451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9820145451\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 41 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 129 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 147 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−17.4324043631, −17.1211991965, −16.6772929038, −15.8582403142, −15.5119037755, −15.0566948824, −14.7513678050, −13.7580880779, −13.4159412967, −12.9537205259, −12.2837006792, −11.6498055040, −11.3368193653, −10.9441342949, −9.98877203020, −9.54753125432, −8.66713293923, −7.88810787093, −7.18895974894, −6.68123501679, −5.92801304442, −4.85985315178, −4.58672818807, −3.56545627221, −2.13577992043,
2.13577992043, 3.56545627221, 4.58672818807, 4.85985315178, 5.92801304442, 6.68123501679, 7.18895974894, 7.88810787093, 8.66713293923, 9.54753125432, 9.98877203020, 10.9441342949, 11.3368193653, 11.6498055040, 12.2837006792, 12.9537205259, 13.4159412967, 13.7580880779, 14.7513678050, 15.0566948824, 15.5119037755, 15.8582403142, 16.6772929038, 17.1211991965, 17.4324043631
