L(s) = 1 | + (1.12 − 2.20i)2-s + (0.885 + 1.48i)3-s + (−2.43 − 3.35i)4-s + (4.28 − 0.281i)6-s + (−0.134 + 0.847i)7-s + (−5.26 + 0.833i)8-s + (−1.43 + 2.63i)9-s + (3.29 − 0.418i)11-s + (2.83 − 6.60i)12-s + (2.14 − 4.20i)13-s + (1.72 + 1.25i)14-s + (−1.52 + 4.67i)16-s + (4.82 − 2.45i)17-s + (4.21 + 6.13i)18-s + (2.92 − 4.03i)19-s + ⋯ |
L(s) = 1 | + (0.796 − 1.56i)2-s + (0.511 + 0.859i)3-s + (−1.21 − 1.67i)4-s + (1.74 − 0.114i)6-s + (−0.0507 + 0.320i)7-s + (−1.86 + 0.294i)8-s + (−0.476 + 0.878i)9-s + (0.992 − 0.126i)11-s + (0.818 − 1.90i)12-s + (0.594 − 1.16i)13-s + (0.459 + 0.334i)14-s + (−0.380 + 1.16i)16-s + (1.17 − 0.596i)17-s + (0.993 + 1.44i)18-s + (0.672 − 0.924i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78771 - 2.07718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78771 - 2.07718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.885 - 1.48i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-3.29 + 0.418i)T \) |
good | 2 | \( 1 + (-1.12 + 2.20i)T + (-1.17 - 1.61i)T^{2} \) |
| 7 | \( 1 + (0.134 - 0.847i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.14 + 4.20i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.82 + 2.45i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 + 4.03i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.93 - 1.93i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.756 - 0.549i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.95 + 6.02i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.46 - 9.22i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.90 - 4.00i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.32 - 2.32i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.85 - 11.6i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (1.46 + 0.748i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.46 + 1.79i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.12 + 9.60i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.03 - 6.03i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.61 - 0.524i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (11.1 + 1.77i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (11.4 - 3.72i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.17 - 12.1i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + (-8.91 - 4.54i)T + (57.0 + 78.4i)T^{2} \) |
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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
−9.914944712797413494604000694485, −9.667080058450720216246479520504, −8.732515305648587615014342197544, −7.66203254355746309755898018131, −5.91211916135463813076189527357, −5.20422256007189882589938870488, −4.23802337598275249054456328257, −3.30901571837458622979854760043, −2.78861807970623219285307621151, −1.21317495251596600434847995347,
1.60153846771326628762794418282, 3.60300931977324353176224413377, 4.00568180137118805523979779075, 5.56061920151381292765436917143, 6.18376173846089046501535063878, 7.08922934170495734271372339829, 7.46433899082398548389503239473, 8.592604575254091143464579181629, 8.977787006343256190134796001230, 10.28094962684569220401132420967