| L(s) = 1 | + (2.08 − 1.20i)2-s + (−0.866 + 1.5i)3-s + (0.910 − 1.57i)4-s + (4.15 − 2.77i)5-s + 4.17i·6-s + (6.98 + 0.420i)7-s + 5.25i·8-s + (−1.5 − 2.59i)9-s + (5.33 − 10.8i)10-s + (6.92 − 11.9i)11-s + (1.57 + 2.73i)12-s − 20.5·13-s + (15.1 − 7.55i)14-s + (0.570 + 8.64i)15-s + (9.98 + 17.2i)16-s + (−6.03 + 10.4i)17-s + ⋯ |
| L(s) = 1 | + (1.04 − 0.603i)2-s + (−0.288 + 0.5i)3-s + (0.227 − 0.394i)4-s + (0.831 − 0.555i)5-s + 0.696i·6-s + (0.998 + 0.0600i)7-s + 0.657i·8-s + (−0.166 − 0.288i)9-s + (0.533 − 1.08i)10-s + (0.629 − 1.08i)11-s + (0.131 + 0.227i)12-s − 1.58·13-s + (1.07 − 0.539i)14-s + (0.0380 + 0.576i)15-s + (0.623 + 1.08i)16-s + (−0.355 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.23353 - 0.442307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23353 - 0.442307i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 + (-4.15 + 2.77i)T \) |
| 7 | \( 1 + (-6.98 - 0.420i)T \) |
| good | 2 | \( 1 + (-2.08 + 1.20i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-6.92 + 11.9i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 20.5T + 169T^{2} \) |
| 17 | \( 1 + (6.03 - 10.4i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10.7 - 6.21i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.5 - 9.56i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 1.74T + 841T^{2} \) |
| 31 | \( 1 + (42.2 + 24.3i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-20.4 + 11.8i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 70.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 45.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (8.98 + 15.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-57.9 - 33.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (30.7 + 17.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-51.9 + 30.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (57.3 + 33.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 31.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-9.99 + 17.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.4 - 85.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 92.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-67.5 + 39.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 93.8T + 9.40e3T^{2} \) |
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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
−13.43140035228425801716037337421, −12.38344079394980909077359722019, −11.56050041537697909467959328845, −10.60319170673786345476930910365, −9.279696848212262616672042318968, −8.132401247007066392004953043303, −5.96153386851633634990625872201, −5.07530617587214826505728335598, −4.01759785779048011944410811397, −2.10453111747791684141050633531,
2.15197933892254093889099263767, 4.51435169738907328935317514355, 5.42443188887645892360149123309, 6.77474299940122593649219874144, 7.38133898863157575141275653605, 9.367265849440353563229430928507, 10.47357087718594471274838336914, 11.88785117175830104058360513530, 12.71923399126713667752862005723, 13.81835028777308309164562940044
