L(s) = 1 | + (−1.57 − 0.573i)3-s + (0.173 + 0.984i)5-s + (0.230 − 0.399i)7-s + (−0.142 − 0.119i)9-s + (−0.642 − 1.11i)11-s + (5.58 − 2.03i)13-s + (0.291 − 1.65i)15-s + (4.26 − 3.57i)17-s + (−3.93 − 1.88i)19-s + (−0.592 + 0.497i)21-s + (0.379 − 2.15i)23-s + (−0.939 + 0.342i)25-s + (2.67 + 4.62i)27-s + (−7.31 − 6.14i)29-s + (4.61 − 8.00i)31-s + ⋯ |
L(s) = 1 | + (−0.910 − 0.331i)3-s + (0.0776 + 0.440i)5-s + (0.0871 − 0.150i)7-s + (−0.0475 − 0.0399i)9-s + (−0.193 − 0.335i)11-s + (1.54 − 0.564i)13-s + (0.0752 − 0.426i)15-s + (1.03 − 0.867i)17-s + (−0.901 − 0.432i)19-s + (−0.129 + 0.108i)21-s + (0.0792 − 0.449i)23-s + (−0.187 + 0.0684i)25-s + (0.514 + 0.890i)27-s + (−1.35 − 1.14i)29-s + (0.829 − 1.43i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.798593 - 0.523376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.798593 - 0.523376i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (3.93 + 1.88i)T \) |
good | 3 | \( 1 + (1.57 + 0.573i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-0.230 + 0.399i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.642 + 1.11i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.58 + 2.03i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.26 + 3.57i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.379 + 2.15i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (7.31 + 6.14i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.61 + 8.00i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 + (-4.53 - 1.64i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.655 + 3.71i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.24 + 4.40i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.37 - 7.78i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (4.10 - 3.44i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.76 - 9.98i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.05 - 4.24i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.19 - 6.78i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (12.3 + 4.47i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.30 + 0.474i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.0660 - 0.114i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.89 + 1.05i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.447 + 0.375i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.22755248585852730496827252601, −10.58361024025011035142788050351, −9.448995785539181532552743861227, −8.282535985081280516390377703754, −7.34056893841116461669672036445, −6.04953438169827061999093911412, −5.82797088063163681080996520787, −4.20462601640062894757055954378, −2.83059485781771187390290782161, −0.78806876298540525708994465675,
1.54881373653986461206402587647, 3.59873599968284434904784867995, 4.76553958477620789278035631177, 5.74664555817116916322746518753, 6.43173796296065897426045853893, 7.957570855465190383204470258122, 8.727148510132278315689486253534, 9.847930689665609341234062788131, 10.81267559550351712096907251454, 11.32237714921758517361042719170