| L(s) = 1 | + (−0.679 + 0.569i)3-s + (−0.939 − 0.342i)5-s + (0.460 − 0.797i)7-s + (−0.384 + 2.18i)9-s + (1.03 + 1.79i)11-s + (3.22 + 2.70i)13-s + (0.833 − 0.303i)15-s + (0.340 + 1.93i)17-s + (0.207 + 4.35i)19-s + (0.141 + 0.803i)21-s + (−0.570 + 0.207i)23-s + (0.766 + 0.642i)25-s + (−2.31 − 4.00i)27-s + (−1.31 + 7.44i)29-s + (1.07 − 1.86i)31-s + ⋯ |
| L(s) = 1 | + (−0.392 + 0.328i)3-s + (−0.420 − 0.152i)5-s + (0.174 − 0.301i)7-s + (−0.128 + 0.726i)9-s + (0.313 + 0.542i)11-s + (0.894 + 0.750i)13-s + (0.215 − 0.0782i)15-s + (0.0825 + 0.468i)17-s + (0.0476 + 0.998i)19-s + (0.0309 + 0.175i)21-s + (−0.118 + 0.0432i)23-s + (0.153 + 0.128i)25-s + (−0.444 − 0.770i)27-s + (−0.243 + 1.38i)29-s + (0.192 − 0.334i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.850917 + 0.636967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850917 + 0.636967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.207 - 4.35i)T \) |
| good | 3 | \( 1 + (0.679 - 0.569i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.460 + 0.797i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 1.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.22 - 2.70i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.340 - 1.93i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.570 - 0.207i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.31 - 7.44i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.07 + 1.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.88T + 37T^{2} \) |
| 41 | \( 1 + (4.50 - 3.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.638 + 0.232i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 10.6i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.25 + 1.54i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.02 + 5.82i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.55 + 2.38i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.32 + 7.52i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (10.8 + 3.93i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.42 + 2.03i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (2.36 - 1.98i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.06 - 8.76i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.60 - 2.18i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (3.29 + 18.7i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.44271802123270396038043012309, −10.71840633295734712143307080810, −9.877613646519752552945285654783, −8.704560954862953622417552026315, −7.88260796180698860642522148538, −6.80557617437231525153818217898, −5.66597810562335628170978957346, −4.56123789008799641978528307797, −3.68152744621448474159355822027, −1.71061334172644079882253470308,
0.808720931616163485852469047015, 2.88692564701645497947256037334, 4.06328552613076766285600188514, 5.53497363675876333637528312296, 6.32428079037144137425954689775, 7.34017566020637240838031451061, 8.431775819383861194043382127150, 9.193642660563882423194589384577, 10.41786964428208208488527162853, 11.48572154704719586131726556355
