| 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
−11.58440242762592156146520754712, −11.00403301748335026840390158335, −9.929094060137093279748654557866, −9.400880874533056766269291643312, −7.84232804632200795877896144950, −6.84535021013106470261214990543, −6.47112213459106153805536437503, −5.51714656824556728219389227221, −2.98016735056284566460980033141, −1.58271005491562845117314599250,
0.839287271657061704877523474365, 2.87414603242709898085306437086, 4.46271296374183471961693385783, 5.77790554944602913937905089761, 6.54583977938947002366380303928, 8.268201388548115835312370455226, 9.141035581418724530239742795582, 9.757853610378703242949468738345, 10.38035376479377106460873038560, 11.52984727670151893627702162248
