| L(s) = 1 | + (−1.33 − 0.470i)2-s + (−1.03 + 1.39i)3-s + (1.55 + 1.25i)4-s + (2.03 − 2.65i)5-s + (2.03 − 1.36i)6-s + (−2.13 + 0.571i)7-s + (−1.48 − 2.40i)8-s + (−0.866 − 2.87i)9-s + (−3.95 + 2.57i)10-s + (1.69 + 0.222i)11-s + (−3.35 + 0.871i)12-s + (−0.378 − 2.87i)13-s + (3.11 + 0.240i)14-s + (1.58 + 5.56i)15-s + (0.855 + 3.90i)16-s + 2.32·17-s + ⋯ |
| L(s) = 1 | + (−0.943 − 0.332i)2-s + (−0.596 + 0.802i)3-s + (0.779 + 0.626i)4-s + (0.909 − 1.18i)5-s + (0.829 − 0.558i)6-s + (−0.806 + 0.216i)7-s + (−0.526 − 0.850i)8-s + (−0.288 − 0.957i)9-s + (−1.25 + 0.815i)10-s + (0.510 + 0.0672i)11-s + (−0.967 + 0.251i)12-s + (−0.105 − 0.798i)13-s + (0.832 + 0.0642i)14-s + (0.409 + 1.43i)15-s + (0.213 + 0.976i)16-s + 0.564·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.634465 - 0.341040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.634465 - 0.341040i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 + 0.470i)T \) |
| 3 | \( 1 + (1.03 - 1.39i)T \) |
| good | 5 | \( 1 + (-2.03 + 2.65i)T + (-1.29 - 4.82i)T^{2} \) |
| 7 | \( 1 + (2.13 - 0.571i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 0.222i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.378 + 2.87i)T + (-12.5 + 3.36i)T^{2} \) |
| 17 | \( 1 - 2.32T + 17T^{2} \) |
| 19 | \( 1 + (-0.0657 + 0.158i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.45 + 9.15i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.01 + 0.782i)T + (7.50 - 28.0i)T^{2} \) |
| 31 | \( 1 + (-4.08 + 2.35i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.93 + 0.801i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-10.1 - 2.72i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.123 - 0.941i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (8.34 + 4.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.49 - 3.51i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (3.18 + 2.44i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-8.42 - 10.9i)T + (-15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (-1.24 - 9.48i)T + (-64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-9.72 + 9.72i)T - 71iT^{2} \) |
| 73 | \( 1 + (5.50 + 5.50i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.56 - 2.71i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.9 - 9.19i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (9.48 + 9.48i)T + 89iT^{2} \) |
| 97 | \( 1 + (-7.16 + 12.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.52984727670151893627702162248, −10.38035376479377106460873038560, −9.757853610378703242949468738345, −9.141035581418724530239742795582, −8.268201388548115835312370455226, −6.54583977938947002366380303928, −5.77790554944602913937905089761, −4.46271296374183471961693385783, −2.87414603242709898085306437086, −0.839287271657061704877523474365,
1.58271005491562845117314599250, 2.98016735056284566460980033141, 5.51714656824556728219389227221, 6.47112213459106153805536437503, 6.84535021013106470261214990543, 7.84232804632200795877896144950, 9.400880874533056766269291643312, 9.929094060137093279748654557866, 11.00403301748335026840390158335, 11.58440242762592156146520754712
