L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 12-s − 2·13-s − 2·14-s + 16-s + 6·17-s + 18-s + 2·21-s − 6·23-s − 24-s − 2·26-s − 27-s − 2·28-s − 4·29-s + 32-s + 6·34-s + 36-s − 10·37-s + 2·39-s − 8·41-s + 2·42-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.436·21-s − 1.25·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.651893597\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651893597\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52266914058985, −13.82405921354343, −13.44965564687058, −12.74995651003036, −12.45268620021172, −11.86419011301350, −11.69999237225222, −10.84673338013684, −10.31154954760831, −9.848904816454004, −9.600596830208349, −8.574599380928929, −8.116409787386586, −7.312091083411218, −7.027867512718625, −6.375224977424368, −5.771799471454366, −5.381182501633216, −4.869540907903958, −3.995623422998989, −3.569068291310010, −3.008364145258202, −2.109589781898219, −1.482328429078991, −0.4001383953293033,
0.4001383953293033, 1.482328429078991, 2.109589781898219, 3.008364145258202, 3.569068291310010, 3.995623422998989, 4.869540907903958, 5.381182501633216, 5.771799471454366, 6.375224977424368, 7.027867512718625, 7.312091083411218, 8.116409787386586, 8.574599380928929, 9.600596830208349, 9.848904816454004, 10.31154954760831, 10.84673338013684, 11.69999237225222, 11.86419011301350, 12.45268620021172, 12.74995651003036, 13.44965564687058, 13.82405921354343, 14.52266914058985