| L(s) = 1 | + (1.79 + 0.380i)2-s + (−1.49 + 0.871i)3-s + (1.23 + 0.551i)4-s + (−0.0204 + 2.23i)5-s + (−3.01 + 0.991i)6-s + (−2.15 + 3.73i)7-s + (−0.953 − 0.692i)8-s + (1.48 − 2.60i)9-s + (−0.888 + 3.99i)10-s + (0.885 + 0.188i)11-s + (−2.33 + 0.253i)12-s + (6.04 − 1.28i)13-s + (−5.29 + 5.87i)14-s + (−1.91 − 3.36i)15-s + (−3.26 − 3.62i)16-s + (4.65 + 3.38i)17-s + ⋯ |
| L(s) = 1 | + (1.26 + 0.269i)2-s + (−0.864 + 0.503i)3-s + (0.619 + 0.275i)4-s + (−0.00914 + 0.999i)5-s + (−1.23 + 0.404i)6-s + (−0.816 + 1.41i)7-s + (−0.337 − 0.244i)8-s + (0.493 − 0.869i)9-s + (−0.280 + 1.26i)10-s + (0.266 + 0.0567i)11-s + (−0.674 + 0.0733i)12-s + (1.67 − 0.356i)13-s + (−1.41 + 1.57i)14-s + (−0.495 − 0.868i)15-s + (−0.815 − 0.905i)16-s + (1.12 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01017 + 1.22454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01017 + 1.22454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.49 - 0.871i)T \) |
| 5 | \( 1 + (0.0204 - 2.23i)T \) |
| good | 2 | \( 1 + (-1.79 - 0.380i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (2.15 - 3.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.885 - 0.188i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-6.04 + 1.28i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-4.65 - 3.38i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.14 + 1.56i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.35 + 2.61i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.238 - 2.26i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.244 + 2.32i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.704 - 2.16i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 1.25i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.16 - 2.00i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.583 + 5.54i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.50 + 2.54i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-12.8 + 2.73i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (13.8 + 2.93i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.796 - 7.57i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (4.21 - 3.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.0260 - 0.0802i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.0932 - 0.887i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.423i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.84 - 5.68i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.171 - 1.63i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.56127473337965738570006098250, −11.76402643031997489085433231596, −10.84279480587660291069927958893, −9.807338170133371482101119654829, −8.700493213083455954035949567874, −6.71732257000589009935331769672, −6.07230780817438044834737312909, −5.53047307258736120499784711964, −3.92131867747127938409265032925, −3.07028475699534538771219895340,
1.11762946934577242663088876464, 3.60183300742592881802237602923, 4.44980498860464769320313272925, 5.65119696274948947437064096873, 6.45052000017065001860880743569, 7.69153775852712221401287296762, 9.109530667796306278230120802393, 10.43204185264518508502072828607, 11.38975138469424284724852361337, 12.19715894476490041318328562373
