L(s) = 1 | + (0.199 − 1.40i)2-s + (−1.65 + 0.520i)3-s + (−1.92 − 0.557i)4-s + (0.400 + 2.41i)6-s + 1.92·7-s + (−1.16 + 2.57i)8-s + (2.45 − 1.72i)9-s − 4.02i·11-s + (3.46 − 0.0792i)12-s − 4.81·13-s + (0.383 − 2.69i)14-s + (3.37 + 2.14i)16-s − 5.23·17-s + (−1.91 − 3.78i)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 + (0.163 + 0.986i)6-s + 0.728·7-s + (−0.411 + 0.911i)8-s + (0.819 − 0.573i)9-s − 1.21i·11-s + (0.999 − 0.0228i)12-s − 1.33·13-s + (0.102 − 0.721i)14-s + (0.844 + 0.535i)16-s − 1.26·17-s + (−0.452 − 0.891i)18-s + 0.157·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0910408 + 0.303981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0910408 + 0.303981i\) |
\(L(\frac{3}{2})\) |
|
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
−10.61516421708753041483891701467, −9.436613465033208324215582561938, −8.747484321244980423855507082832, −7.50834790974509333067700897129, −6.22484398779042838638387635648, −5.17760027386642915193496991941, −4.60848720226734668752490198907, −3.38686326198585277129252365697, −1.86809360848368401651626968453, −0.18540999638234605428406396590,
1.98247445607578083518216549422, 4.22205362170754678099261303652, 4.91359822285350098898384625881, 5.63929207260788845882185783914, 7.03336227947288117921811540995, 7.17427806018952847415125237697, 8.263954643311309201848759486867, 9.472961095954304797037329466101, 10.15360562212321142988512654758, 11.34364208860415241107026039737