| L(s) = 1 | + (−1.34 + 0.452i)2-s + (0.961 + 1.44i)3-s + (1.59 − 1.21i)4-s + (−1.07 − 0.527i)5-s + (−1.93 − 1.49i)6-s + (−2.72 − 3.55i)7-s + (−1.58 + 2.34i)8-s + (−1.15 + 2.77i)9-s + (1.67 + 0.223i)10-s + (1.76 + 1.54i)11-s + (3.27 + 1.12i)12-s + (−0.342 − 0.0224i)13-s + (5.25 + 3.52i)14-s + (−0.268 − 2.05i)15-s + (1.06 − 3.85i)16-s + (−4.44 − 4.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.947 + 0.319i)2-s + (0.555 + 0.831i)3-s + (0.795 − 0.605i)4-s + (−0.478 − 0.236i)5-s + (−0.791 − 0.610i)6-s + (−1.02 − 1.34i)7-s + (−0.560 + 0.828i)8-s + (−0.383 + 0.923i)9-s + (0.529 + 0.0706i)10-s + (0.532 + 0.466i)11-s + (0.945 + 0.325i)12-s + (−0.0949 − 0.00622i)13-s + (1.40 + 0.942i)14-s + (−0.0693 − 0.529i)15-s + (0.266 − 0.963i)16-s + (−1.07 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.400923 - 0.337068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400923 - 0.337068i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 0.452i)T \) |
| 3 | \( 1 + (-0.961 - 1.44i)T \) |
| good | 5 | \( 1 + (1.07 + 0.527i)T + (3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (2.72 + 3.55i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (-1.76 - 1.54i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.342 + 0.0224i)T + (12.8 + 1.69i)T^{2} \) |
| 17 | \( 1 + (4.44 + 4.44i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.08 + 3.12i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (1.36 - 1.77i)T + (-5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-2.36 + 6.97i)T + (-23.0 - 17.6i)T^{2} \) |
| 31 | \( 1 + (3.16 + 5.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.06 + 1.38i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-9.02 - 6.92i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (6.56 + 5.76i)T + (5.61 + 42.6i)T^{2} \) |
| 47 | \( 1 + (2.15 + 8.04i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.173 + 0.0345i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (5.42 - 11.0i)T + (-35.9 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 4.78i)T + (-48.3 - 37.1i)T^{2} \) |
| 67 | \( 1 + (-1.23 + 1.08i)T + (8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (12.5 + 5.20i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-6.08 + 2.52i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.01 - 7.53i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.223 + 0.110i)T + (50.5 - 65.8i)T^{2} \) |
| 89 | \( 1 + (0.644 + 0.266i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 5.81i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.15220844822188818088836612959, −9.613864930511100742145774198811, −9.030267857572634986999302559665, −7.82834269760921528284242084016, −7.23879834427548192984127908005, −6.26849968452713335689847840559, −4.71205820622461511472245137392, −3.83860124118634413793349660126, −2.50997544452103057654975454251, −0.36064818420270861461940704947,
1.69412161055821157091962400047, 2.90260161423550275732804784042, 3.63171067590896354037124270118, 5.99399787634469451029647609859, 6.53616163698333742785241444295, 7.51712879880475791786133685831, 8.559445509492158724741924187803, 8.923644750144794129455180757461, 9.779916539803952811460798067503, 10.98980555207161934432665208914
