L(s) = 1 | + (−0.199 + 1.40i)2-s + (1.65 + 0.520i)3-s + (−1.92 − 0.557i)4-s + (−1.05 + 2.20i)6-s + 1.92·7-s + (1.16 − 2.57i)8-s + (2.45 + 1.72i)9-s + 4.02i·11-s + (−2.88 − 1.92i)12-s − 4.81·13-s + (−0.383 + 2.69i)14-s + (3.37 + 2.14i)16-s + 5.23·17-s + (−2.89 + 3.09i)18-s + 0.684·19-s + ⋯ |
L(s) = 1 | + (−0.140 + 0.990i)2-s + (0.953 + 0.300i)3-s + (−0.960 − 0.278i)4-s + (−0.431 + 0.901i)6-s + 0.728·7-s + (0.411 − 0.911i)8-s + (0.819 + 0.573i)9-s + 1.21i·11-s + (−0.832 − 0.554i)12-s − 1.33·13-s + (−0.102 + 0.721i)14-s + (0.844 + 0.535i)16-s + 1.26·17-s + (−0.682 + 0.730i)18-s + 0.157·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01711 + 1.49858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01711 + 1.49858i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.199 - 1.40i)T \) |
| 3 | \( 1 + (-1.65 - 0.520i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.92T + 7T^{2} \) |
| 11 | \( 1 - 4.02iT - 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 0.684T + 19T^{2} \) |
| 23 | \( 1 - 1.72iT - 23T^{2} \) |
| 29 | \( 1 - 6.99T + 29T^{2} \) |
| 31 | \( 1 - 4.23iT - 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 - 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 1.04iT - 43T^{2} \) |
| 47 | \( 1 + 7.55iT - 47T^{2} \) |
| 53 | \( 1 + 4.08iT - 53T^{2} \) |
| 59 | \( 1 - 0.994iT - 59T^{2} \) |
| 61 | \( 1 + 3.16iT - 61T^{2} \) |
| 67 | \( 1 + 14.8iT - 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 9.25iT - 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 + 0.829iT - 89T^{2} \) |
| 97 | \( 1 + 1.45iT - 97T^{2} \) |
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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
−10.32308543504864918279732239238, −9.948973959123671050201650149655, −9.085038415143154793125978527638, −8.110421657821254333214257475313, −7.53922331798408186091239931884, −6.80670499415829296571067941764, −5.07390630439101133507835954953, −4.77949093484879256050597904982, −3.40016496742661654193378157660, −1.78255500560737514818526682853,
1.09116702674477044598071291462, 2.46245770550679564616647207401, 3.31813354179807653832964626409, 4.47602556885068564856932009746, 5.55428769939001080860465935750, 7.19351862831904719835471160544, 8.109791898981119944281643615921, 8.579642295951260844388192048594, 9.644893569272424393705382567961, 10.25819885619790027824607645693