L(s) = 1 | + (1.22 + 1.22i)3-s + (6.45 + 6.45i)5-s + 8.74·7-s + 2.99i·9-s + (−9.12 + 9.12i)11-s + (−9.18 + 9.18i)13-s + 15.8i·15-s + 18.9·17-s + (−2.79 − 2.79i)19-s + (10.7 + 10.7i)21-s − 2.27·23-s + 58.2i·25-s + (−3.67 + 3.67i)27-s + (−10.8 + 10.8i)29-s − 50.9i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (1.29 + 1.29i)5-s + 1.24·7-s + 0.333i·9-s + (−0.829 + 0.829i)11-s + (−0.706 + 0.706i)13-s + 1.05i·15-s + 1.11·17-s + (−0.147 − 0.147i)19-s + (0.510 + 0.510i)21-s − 0.0988·23-s + 2.33i·25-s + (−0.136 + 0.136i)27-s + (−0.375 + 0.375i)29-s − 1.64i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.887903395\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.887903395\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
good | 5 | \( 1 + (-6.45 - 6.45i)T + 25iT^{2} \) |
| 7 | \( 1 - 8.74T + 49T^{2} \) |
| 11 | \( 1 + (9.12 - 9.12i)T - 121iT^{2} \) |
| 13 | \( 1 + (9.18 - 9.18i)T - 169iT^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 19 | \( 1 + (2.79 + 2.79i)T + 361iT^{2} \) |
| 23 | \( 1 + 2.27T + 529T^{2} \) |
| 29 | \( 1 + (10.8 - 10.8i)T - 841iT^{2} \) |
| 31 | \( 1 + 50.9iT - 961T^{2} \) |
| 37 | \( 1 + (47.2 + 47.2i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 27.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-35.9 + 35.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 61.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-53.7 - 53.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-20.5 + 20.5i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (12.6 - 12.6i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-43.7 - 43.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 20.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 49.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-15.0 - 15.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 62.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.7T + 9.40e3T^{2} \) |
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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
−10.26637114205224707513269809627, −9.732866634331685059750085491684, −8.794942097468747374191632213093, −7.49362720353009851407218250741, −7.18929712497739237748456620230, −5.71977098204121297709523295260, −5.12951925928796644576992184442, −3.86344804255833405206970217712, −2.39415879313893392388628029868, −2.00388707756329164752344059211,
0.960416331598200766765460910486, 1.84279137327013258189192725285, 3.04291107184139490904888745349, 4.84019318638446446428065850928, 5.27543268925400308308997332704, 6.12931562898639745171030549472, 7.64913655731850814983983572136, 8.239280554314292364686901443700, 8.846636258105844364008997536668, 9.890160362499108881451363427438