| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·7-s − 3·8-s − 2·9-s − 10-s + 11-s − 12-s + 13-s − 3·14-s − 15-s − 16-s − 3·17-s − 2·18-s + 7·19-s + 20-s − 3·21-s + 22-s + 5·23-s − 3·24-s − 4·25-s + 26-s − 5·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.258·15-s − 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.04·23-s − 0.612·24-s − 4/5·25-s + 0.196·26-s − 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9456327855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9456327855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.61177941895200, −14.95636781825731, −14.52667101251383, −13.87526338716702, −13.48035810695165, −13.17472347543430, −12.35709576264269, −12.06294109275533, −11.29668700459066, −10.86137560020418, −9.807194984947151, −9.385634706116369, −9.004605964227371, −8.506455478326069, −7.651057135281372, −7.119936826623442, −6.388264978041982, −5.640386059969361, −5.337730743473348, −4.335521730432193, −3.751244812525411, −3.165176125545778, −2.913871713159112, −1.660117468916551, −0.3368993521046619,
0.3368993521046619, 1.660117468916551, 2.913871713159112, 3.165176125545778, 3.751244812525411, 4.335521730432193, 5.337730743473348, 5.640386059969361, 6.388264978041982, 7.119936826623442, 7.651057135281372, 8.506455478326069, 9.004605964227371, 9.385634706116369, 9.807194984947151, 10.86137560020418, 11.29668700459066, 12.06294109275533, 12.35709576264269, 13.17472347543430, 13.48035810695165, 13.87526338716702, 14.52667101251383, 14.95636781825731, 15.61177941895200
