| L(s) = 1 | + (1.28 − 0.598i)2-s + (−0.161 − 1.72i)3-s + (1.28 − 1.53i)4-s + (0.0476 − 0.817i)5-s + (−1.23 − 2.11i)6-s + (0.895 + 3.77i)7-s + (0.724 − 2.73i)8-s + (−2.94 + 0.556i)9-s + (−0.428 − 1.07i)10-s + (−2.10 − 3.19i)11-s + (−2.85 − 1.96i)12-s + (−0.800 − 6.85i)13-s + (3.40 + 4.30i)14-s + (−1.41 + 0.0497i)15-s + (−0.709 − 3.93i)16-s + (−1.41 + 1.68i)17-s + ⋯ |
| L(s) = 1 | + (0.905 − 0.423i)2-s + (−0.0931 − 0.995i)3-s + (0.641 − 0.767i)4-s + (0.0212 − 0.365i)5-s + (−0.506 − 0.862i)6-s + (0.338 + 1.42i)7-s + (0.256 − 0.966i)8-s + (−0.982 + 0.185i)9-s + (−0.135 − 0.340i)10-s + (−0.633 − 0.963i)11-s + (−0.823 − 0.567i)12-s + (−0.222 − 1.89i)13-s + (0.911 + 1.15i)14-s + (−0.366 + 0.0128i)15-s + (−0.177 − 0.984i)16-s + (−0.342 + 0.408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00552 - 2.10882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00552 - 2.10882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 + 0.598i)T \) |
| 3 | \( 1 + (0.161 + 1.72i)T \) |
| good | 5 | \( 1 + (-0.0476 + 0.817i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (-0.895 - 3.77i)T + (-6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (2.10 + 3.19i)T + (-4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.800 + 6.85i)T + (-12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (1.41 - 1.68i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.21 + 2.69i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 0.331i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (5.67 - 7.61i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (-5.07 + 1.52i)T + (25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.02 - 0.181i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-3.49 - 1.50i)T + (28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-4.06 - 2.04i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-0.178 + 0.597i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (4.47 - 7.74i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.85 + 4.34i)T + (-23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (5.51 - 5.20i)T + (3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-5.20 - 6.99i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (-13.6 + 4.97i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.82 - 2.84i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.11 + 1.77i)T + (54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (1.64 - 0.711i)T + (56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (2.66 - 7.33i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.362 - 6.22i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−10.77838242833908300570762879763, −9.294242333411036287916296507752, −8.389439083829961602220943153166, −7.60328576691270651108839714976, −6.32991224825444196729234636964, −5.37615229919190563985872030450, −5.25794873413104566116081889745, −3.10212857043027327545476203059, −2.55530300214391426432537225081, −0.989081531445695328416974764338,
2.30222108204936861807408445625, 3.72615129010327312489204323490, 4.42465189912839278737213337731, 5.03468446879948530401015801421, 6.42920057386816326264071075424, 7.17921710260680648271660068252, 7.948369658648481634730382351566, 9.316938241380836582555767066963, 10.10170459476770316236757216233, 11.04031105190940026628841246749
