L(s) = 1 | + (−1.58 + 1.21i)2-s + (1.03 − 3.86i)4-s + 1.46·5-s − 6.38i·7-s + (3.06 + 7.39i)8-s + (−2.32 + 1.78i)10-s − 9.33i·11-s − 22.7·13-s + (7.77 + 10.1i)14-s + (−13.8 − 8.00i)16-s − 5.17·17-s − 4.35i·19-s + (1.52 − 5.67i)20-s + (11.3 + 14.8i)22-s + 37.0i·23-s + ⋯ |
L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + 0.293·5-s − 0.912i·7-s + (0.382 + 0.923i)8-s + (−0.232 + 0.178i)10-s − 0.849i·11-s − 1.74·13-s + (0.555 + 0.724i)14-s + (−0.866 − 0.500i)16-s − 0.304·17-s − 0.229i·19-s + (0.0760 − 0.283i)20-s + (0.516 + 0.673i)22-s + 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1948192171\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1948192171\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.58 - 1.21i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 - 1.46T + 25T^{2} \) |
| 7 | \( 1 + 6.38iT - 49T^{2} \) |
| 11 | \( 1 + 9.33iT - 121T^{2} \) |
| 13 | \( 1 + 22.7T + 169T^{2} \) |
| 17 | \( 1 + 5.17T + 289T^{2} \) |
| 23 | \( 1 - 37.0iT - 529T^{2} \) |
| 29 | \( 1 - 23.0T + 841T^{2} \) |
| 31 | \( 1 - 19.9iT - 961T^{2} \) |
| 37 | \( 1 - 24.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 78.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 86.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 114.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 64.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 58.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 47.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 51.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 4.37T + 7.92e3T^{2} \) |
| 97 | \( 1 - 16.1T + 9.40e3T^{2} \) |
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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
−10.38503515049668232288865913465, −9.767666165084071605680970032233, −9.034898709445464702902429997723, −7.86424217601747321547616429909, −7.36472231856012591422029926489, −6.41364314422782494666483666300, −5.43609125643502683156416688802, −4.46414318101830850451733247724, −2.83538442516177824967623264574, −1.34732316129551013934105171865,
0.089552896539637301321629034116, 2.08877072315929092942131885891, 2.56947873375738428582583866479, 4.19120761523446376612240633485, 5.22290035928776300505558990979, 6.57651631810090441070895219602, 7.38140852177736904167374749437, 8.367230150471796474240964555476, 9.132215000171452575026500377223, 9.998743263294264973739514698685