L(s) = 1 | + (−0.639 − 1.26i)2-s + (0.730 − 1.57i)3-s + (−1.18 + 1.61i)4-s + (−2.44 + 0.0838i)6-s + 1.25·7-s + (2.79 + 0.458i)8-s + (−1.93 − 2.29i)9-s − 3.02i·11-s + (1.67 + 3.03i)12-s + 5.65·13-s + (−0.803 − 1.58i)14-s + (−1.20 − 3.81i)16-s − 2.45·17-s + (−1.65 + 3.90i)18-s − 1.77·19-s + ⋯ |
L(s) = 1 | + (−0.452 − 0.891i)2-s + (0.421 − 0.906i)3-s + (−0.590 + 0.806i)4-s + (−0.999 + 0.0342i)6-s + 0.474·7-s + (0.986 + 0.162i)8-s + (−0.644 − 0.764i)9-s − 0.911i·11-s + (0.482 + 0.875i)12-s + 1.56·13-s + (−0.214 − 0.423i)14-s + (−0.301 − 0.953i)16-s − 0.595·17-s + (−0.390 + 0.920i)18-s − 0.407·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.316964 - 1.21766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.316964 - 1.21766i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.639 + 1.26i)T \) |
| 3 | \( 1 + (-0.730 + 1.57i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 3.02iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.45T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 8.84iT - 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 + 7.57iT - 41T^{2} \) |
| 43 | \( 1 + 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 1.83iT - 47T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.91iT - 59T^{2} \) |
| 61 | \( 1 - 8.16iT - 61T^{2} \) |
| 67 | \( 1 + 8.50iT - 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 7.36iT - 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.70743002092579566054407517840, −9.099650527953389759682200737913, −8.609124450127124343189146958142, −8.050656046612726678823518403787, −6.85981035302807935315888607578, −5.87155830938912153955541970085, −4.27324472314500786472253643841, −3.22740971820826627958428338046, −2.08156887077094274928390909856, −0.831464145959104830062816294881,
1.79123396163390379237861074789, 3.72029965112689569914964382790, 4.59529799673671755304315808603, 5.55565600519165672367253691034, 6.53662976744032747103032054710, 7.82573237445811701237071643451, 8.275008643005160368593978650457, 9.429889927088677463178530031162, 9.685384292630163793236684238501, 11.02301620999507313177517595118