| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·5-s − 2·6-s + 2·7-s − 8-s + 9-s + 2·10-s − 5·11-s + 2·12-s − 2·13-s − 2·14-s − 4·15-s + 16-s + 4·17-s − 18-s − 6·19-s − 2·20-s + 4·21-s + 5·22-s + 6·23-s − 2·24-s − 25-s + 2·26-s − 4·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.50·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.447·20-s + 0.872·21-s + 1.06·22-s + 1.25·23-s − 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473483029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473483029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3023 | \( 1+O(T) \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.055812111845463618111878037244, −7.72458810360434144423121242344, −7.16654216871008798508129946907, −6.01718701201089838330447618078, −5.08771144529883456245182137736, −4.36429121603306603378300811796, −3.37192102463411052752234160869, −2.66456315487841522626642311979, −2.04706243461322298082259195106, −0.64325620817188795056064808324,
0.64325620817188795056064808324, 2.04706243461322298082259195106, 2.66456315487841522626642311979, 3.37192102463411052752234160869, 4.36429121603306603378300811796, 5.08771144529883456245182137736, 6.01718701201089838330447618078, 7.16654216871008798508129946907, 7.72458810360434144423121242344, 8.055812111845463618111878037244
