| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s − 13-s − 2·14-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 2·21-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 2·28-s − 30-s − 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.436·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.182·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7009906798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7009906798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.82648535001870, −14.11749402940528, −13.64395275205747, −12.75610226240250, −12.42374257106271, −12.00231303462489, −11.22453430736283, −10.98531755785550, −10.67782530913186, −9.801216488186431, −9.494150152785768, −8.657206690259877, −8.302336152497766, −7.773976587281122, −7.159809657321918, −6.693766163400348, −6.080131734243036, −5.344844499374147, −4.928063108844077, −3.992214754084135, −3.796593826226240, −2.494751523173121, −2.096875452352682, −1.246832375671593, −0.3652765200048942,
0.3652765200048942, 1.246832375671593, 2.096875452352682, 2.494751523173121, 3.796593826226240, 3.992214754084135, 4.928063108844077, 5.344844499374147, 6.080131734243036, 6.693766163400348, 7.159809657321918, 7.773976587281122, 8.302336152497766, 8.657206690259877, 9.494150152785768, 9.801216488186431, 10.67782530913186, 10.98531755785550, 11.22453430736283, 12.00231303462489, 12.42374257106271, 12.75610226240250, 13.64395275205747, 14.11749402940528, 14.82648535001870
