L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.22 + 2.12i)5-s + (−1 + 2.44i)7-s + 0.999i·8-s + (−2.12 + 1.22i)10-s + (−3.67 + 2.12i)11-s − 4.18i·13-s + (−2.09 + 1.62i)14-s + (−0.5 + 0.866i)16-s + (−1.22 − 2.12i)17-s + (4.24 + 2.44i)19-s − 2.44·20-s − 4.24·22-s + (5.19 + 3i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.547 + 0.948i)5-s + (−0.377 + 0.925i)7-s + 0.353i·8-s + (−0.670 + 0.387i)10-s + (−1.10 + 0.639i)11-s − 1.15i·13-s + (−0.558 + 0.433i)14-s + (−0.125 + 0.216i)16-s + (−0.297 − 0.514i)17-s + (0.973 + 0.561i)19-s − 0.547·20-s − 0.904·22-s + (1.08 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.725775 + 1.27061i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.725775 + 1.27061i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 5 | \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.18iT - 13T^{2} \) |
| 17 | \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.24 - 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + (-3.97 + 6.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.15 + 1.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 2.12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.621 + 0.358i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 3.01i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (13.2 - 7.64i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.62 - 8.00i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.49T + 83T^{2} \) |
| 89 | \( 1 + (-8.87 + 15.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.63iT - 97T^{2} \) |
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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
−11.77091299779775380806505818143, −10.86954330708787085577898820478, −9.997297079003790759395636516219, −8.757699378618956612764061741524, −7.57190389867283249525724145430, −7.10180871557025749508183412722, −5.73122122292843778804414798441, −5.04028860507216640092781304137, −3.33064832592204535138309223376, −2.70371834603069729774990166611,
0.824009368639995085470008281421, 2.81855434303915579719796488249, 4.18672145210840021439404454344, 4.79968519774219734071919844863, 6.12783474940424461333674694372, 7.24677746307212328023735942723, 8.266292611141300835988971129589, 9.287233216963825749759262131074, 10.33614874097371869673844274233, 11.20081687807881202731732554808