L(s) = 1 | + (−0.5 − 0.866i)3-s + 2.67·5-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.615 + 1.06i)11-s + (−3.44 − 1.06i)13-s + (−1.33 − 2.31i)15-s + (−3.68 + 6.39i)17-s + (−3.11 + 5.39i)19-s − 0.999·21-s + (2.85 + 4.93i)23-s + 2.16·25-s + 0.999·27-s + (4.52 + 7.84i)29-s + 2.10·31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + 1.19·5-s + (0.188 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.185 + 0.321i)11-s + (−0.955 − 0.295i)13-s + (−0.345 − 0.598i)15-s + (−0.894 + 1.54i)17-s + (−0.714 + 1.23i)19-s − 0.218·21-s + (0.594 + 1.02i)23-s + 0.433·25-s + 0.192·27-s + (0.841 + 1.45i)29-s + 0.377·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.568235750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.568235750\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (3.44 + 1.06i)T \) |
good | 5 | \( 1 - 2.67T + 5T^{2} \) |
| 11 | \( 1 + (-0.615 - 1.06i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.68 - 6.39i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 - 5.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.85 - 4.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.52 - 7.84i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.10T + 31T^{2} \) |
| 37 | \( 1 + (-4.01 - 6.95i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.27 - 5.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.08 + 5.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 + (-6.69 + 11.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.71 - 8.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.12 + 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.68 + 11.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 - 2.08T + 83T^{2} \) |
| 89 | \( 1 + (-0.240 - 0.417i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.17 - 3.76i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.279312940195060460825050066730, −8.303197894320977790586354306287, −7.68569510259060456246425308162, −6.51792648363892564968469011151, −6.31932163698220146542517339060, −5.26250338999276433297128835621, −4.55536289008974521276540459029, −3.28343572095092704027739959095, −2.02005005254339481907472198354, −1.45719809284917731037625351869,
0.54414520450282894527697164154, 2.35786709029187669348985487189, 2.67017158902506041811671647863, 4.47590915925600902227366171147, 4.78404202261729018575873178292, 5.78732935585625339010957778335, 6.49531364410896612841903151522, 7.16612956094755094484742603669, 8.410725691392683118348450366410, 9.220890495817479310476253371548