L(s) = 1 | + (−1.80 + 0.863i)2-s + (−1.22 − 1.22i)3-s + (2.50 − 3.11i)4-s + (6.49 + 6.49i)5-s + (3.26 + 1.15i)6-s + 3.94·7-s + (−1.83 + 7.78i)8-s + 2.99i·9-s + (−17.3 − 6.10i)10-s + (4.31 − 4.31i)11-s + (−6.88 + 0.743i)12-s + (4.06 − 4.06i)13-s + (−7.11 + 3.40i)14-s − 15.9i·15-s + (−3.41 − 15.6i)16-s − 14.5·17-s + ⋯ |
L(s) = 1 | + (−0.901 + 0.431i)2-s + (−0.408 − 0.408i)3-s + (0.627 − 0.778i)4-s + (1.29 + 1.29i)5-s + (0.544 + 0.191i)6-s + 0.563·7-s + (−0.229 + 0.973i)8-s + 0.333i·9-s + (−1.73 − 0.610i)10-s + (0.391 − 0.391i)11-s + (−0.574 + 0.0619i)12-s + (0.312 − 0.312i)13-s + (−0.508 + 0.243i)14-s − 1.06i·15-s + (−0.213 − 0.976i)16-s − 0.856·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.802682 + 0.251676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.802682 + 0.251676i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.80 - 0.863i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-6.49 - 6.49i)T + 25iT^{2} \) |
| 7 | \( 1 - 3.94T + 49T^{2} \) |
| 11 | \( 1 + (-4.31 + 4.31i)T - 121iT^{2} \) |
| 13 | \( 1 + (-4.06 + 4.06i)T - 169iT^{2} \) |
| 17 | \( 1 + 14.5T + 289T^{2} \) |
| 19 | \( 1 + (-4.94 - 4.94i)T + 361iT^{2} \) |
| 23 | \( 1 + 43.6T + 529T^{2} \) |
| 29 | \( 1 + (-25.0 + 25.0i)T - 841iT^{2} \) |
| 31 | \( 1 + 32.5iT - 961T^{2} \) |
| 37 | \( 1 + (-4.14 - 4.14i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 55.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (16.1 - 16.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 7.92iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (31.5 + 31.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (49.7 - 49.7i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-44.4 + 44.4i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (1.64 + 1.64i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 10.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 72.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-42.0 - 42.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 28.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 54.2T + 9.40e3T^{2} \) |
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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
−15.63330350274993862752635468238, −14.37308113958754457893814294537, −13.68673348488977545754787644734, −11.58056419200418922680884135427, −10.64957327401779007875698372688, −9.675175312592255847189541040831, −8.019604165540918242977519368097, −6.57794238394518717753423782922, −5.83486076223047143544175054607, −2.09422229771337631941299255751,
1.64318431323729946503727064285, 4.65808041655909513948753146786, 6.34446829505170908899301860358, 8.437509926924430584523506485421, 9.357094070571521104669569021504, 10.31247069241421869876252357018, 11.71717206419249596510768051600, 12.72730822416375301608100705845, 14.02190501437789231329548190215, 15.89559665767867146524215430812