| L(s) = 1 | + (−1.06 − 0.928i)2-s + (1.59 + 2.19i)3-s + (0.275 + 1.98i)4-s + (−0.720 − 2.21i)5-s + (0.337 − 3.83i)6-s + (−1.04 − 0.758i)7-s + (1.54 − 2.36i)8-s + (−1.35 + 4.17i)9-s + (−1.29 + 3.03i)10-s + (−3.29 − 0.387i)11-s + (−3.91 + 3.77i)12-s + (−0.279 − 0.0906i)13-s + (0.409 + 1.77i)14-s + (3.72 − 5.13i)15-s + (−3.84 + 1.09i)16-s + (2.82 − 0.917i)17-s + ⋯ |
| L(s) = 1 | + (−0.754 − 0.656i)2-s + (0.922 + 1.26i)3-s + (0.137 + 0.990i)4-s + (−0.322 − 0.992i)5-s + (0.137 − 1.56i)6-s + (−0.394 − 0.286i)7-s + (0.546 − 0.837i)8-s + (−0.452 + 1.39i)9-s + (−0.408 + 0.960i)10-s + (−0.993 − 0.116i)11-s + (−1.13 + 1.08i)12-s + (−0.0774 − 0.0251i)13-s + (0.109 + 0.475i)14-s + (0.962 − 1.32i)15-s + (−0.962 + 0.272i)16-s + (0.684 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678040 - 0.000342550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678040 - 0.000342550i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.06 + 0.928i)T \) |
| 11 | \( 1 + (3.29 + 0.387i)T \) |
| good | 3 | \( 1 + (-1.59 - 2.19i)T + (-0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (0.720 + 2.21i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.04 + 0.758i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (0.279 + 0.0906i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 0.917i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.38 - 1.00i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 0.525iT - 23T^{2} \) |
| 29 | \( 1 + (4.84 - 6.66i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.22 - 1.37i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.22 - 3.07i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.28 - 4.52i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 + (4.50 + 6.19i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.484 + 1.49i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.27 - 11.3i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.98 - 2.91i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 + (-3.41 + 1.11i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.51 + 3.46i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.04 + 9.36i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.16 + 3.57i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 0.598T + 89T^{2} \) |
| 97 | \( 1 + (2.57 - 7.91i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−16.32378355053850877660516075545, −15.09422346323137474356556342537, −13.48157075451945699382504459317, −12.37527871665517424063079528518, −10.72626767257690258262512858420, −9.801604001178740238351723333664, −8.809652257184496275931719876063, −7.85386223610384049662511390680, −4.65681933478518202795766550077, −3.20137025099284401050165051776,
2.58412155340676958706680891206, 6.16118337944867198539783465946, 7.39060528348956720865968018095, 8.050575287476716321024773099371, 9.559624870350844319509062624985, 11.01224090524692704958845284124, 12.70080500588502206200436874215, 13.90062962978395846363859517661, 14.84240553543254265135120661442, 15.71684503145582864792387485310
