L(s) = 1 | + (−0.185 − 0.693i)2-s + i·3-s + (1.28 − 0.741i)4-s + (0.295 − 1.10i)5-s + (0.693 − 0.185i)6-s + (−0.650 − 2.56i)7-s + (−1.76 − 1.76i)8-s − 9-s − 0.818·10-s + (−1.26 − 1.26i)11-s + (0.741 + 1.28i)12-s + (3.60 + 0.189i)13-s + (−1.65 + 0.928i)14-s + (1.10 + 0.295i)15-s + (0.584 − 1.01i)16-s + (1.84 + 3.19i)17-s + ⋯ |
L(s) = 1 | + (−0.131 − 0.490i)2-s + 0.577i·3-s + (0.642 − 0.370i)4-s + (0.131 − 0.492i)5-s + (0.283 − 0.0759i)6-s + (−0.245 − 0.969i)7-s + (−0.625 − 0.625i)8-s − 0.333·9-s − 0.258·10-s + (−0.381 − 0.381i)11-s + (0.214 + 0.370i)12-s + (0.998 + 0.0525i)13-s + (−0.443 + 0.248i)14-s + (0.284 + 0.0761i)15-s + (0.146 − 0.253i)16-s + (0.447 + 0.775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07790 - 0.797415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07790 - 0.797415i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.650 + 2.56i)T \) |
| 13 | \( 1 + (-3.60 - 0.189i)T \) |
good | 2 | \( 1 + (0.185 + 0.693i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.295 + 1.10i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.26 + 1.26i)T + 11iT^{2} \) |
| 17 | \( 1 + (-1.84 - 3.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.439 + 0.439i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.96 + 1.13i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0743 + 0.128i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.64 + 1.24i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (4.33 - 1.16i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.12 - 4.19i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.81 - 3.93i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.291 + 0.0779i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-13.7 - 3.68i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 2.90iT - 61T^{2} \) |
| 67 | \( 1 + (8.53 - 8.53i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.67 + 13.6i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.05 - 11.3i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.537 + 0.930i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.90 + 3.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.99 - 11.1i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (4.98 - 1.33i)T + (84.0 - 48.5i)T^{2} \) |
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
−11.42755061081281059278005314946, −10.65139039063727706776020014074, −10.13244551724879638611320884542, −9.063667735227283919382589193791, −7.966886168319505424104343526762, −6.59275504241109400906047792203, −5.68039334446493268208129126775, −4.21255181893280983935577818527, −3.07518290684097304793618237858, −1.18339853431145674414335189055,
2.20789022817692179609874099996, 3.24762051809016545702593635062, 5.41421832976408007061326332672, 6.31355351633346900068134116748, 7.09311711129503876103368870571, 8.121969602996287491168726336864, 8.915141219575802357067347007967, 10.26699454352348636704012403834, 11.35455043734239547451233110779, 12.08591597425992256335012122543