L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 12-s + 4·13-s + 4·14-s + 16-s + 6·17-s − 18-s + 4·21-s − 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s − 4·28-s − 6·29-s − 2·31-s − 32-s − 6·34-s + 36-s + 4·37-s − 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.872·21-s − 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.657·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7079648619\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7079648619\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.259640558821885075068908771795, −8.286384627590355256677817927703, −7.55115786695816520855805315822, −6.68048915127927837981186032967, −5.99408682820686103023132488037, −5.52310079772540467707832315072, −3.85774434155809618478109209426, −3.38136823890594312411975627313, −1.95377788914540392982009269037, −0.61960001960809225980571440734,
0.61960001960809225980571440734, 1.95377788914540392982009269037, 3.38136823890594312411975627313, 3.85774434155809618478109209426, 5.52310079772540467707832315072, 5.99408682820686103023132488037, 6.68048915127927837981186032967, 7.55115786695816520855805315822, 8.286384627590355256677817927703, 9.259640558821885075068908771795