L(s) = 1 | + (0.766 + 0.642i)2-s + (1.26 − 0.460i)3-s + (0.173 + 0.984i)4-s + (1.26 + 0.460i)6-s + (1.43 + 2.49i)7-s + (−0.500 + 0.866i)8-s + (−0.907 + 0.761i)9-s + (0.5 − 0.866i)11-s + (0.673 + 1.16i)12-s + (0.5 + 0.181i)13-s + (−0.500 + 2.83i)14-s + (−0.939 + 0.342i)16-s + (1.26 + 1.06i)17-s − 1.18·18-s + (3.79 + 2.15i)19-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (0.730 − 0.266i)3-s + (0.0868 + 0.492i)4-s + (0.516 + 0.188i)6-s + (0.544 + 0.942i)7-s + (−0.176 + 0.306i)8-s + (−0.302 + 0.253i)9-s + (0.150 − 0.261i)11-s + (0.194 + 0.336i)12-s + (0.138 + 0.0504i)13-s + (−0.133 + 0.757i)14-s + (−0.234 + 0.0855i)16-s + (0.307 + 0.257i)17-s − 0.279·18-s + (0.869 + 0.493i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.25179 + 1.61480i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25179 + 1.61480i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.79 - 2.15i)T \) |
good | 3 | \( 1 + (-1.26 + 0.460i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.181i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 1.06i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.194 + 1.10i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.81 - 1.52i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.847 - 1.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 + (4.85 - 1.76i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.88 + 10.6i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.39 + 1.16i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.08 + 11.8i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.72 - 5.64i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 6.15i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.93 + 1.62i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.02 - 5.79i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-4.68 + 1.70i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-8.05 + 2.93i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.32 + 9.22i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.694 + 0.252i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.88 + 7.45i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.12033376613814373024054849004, −8.970847976258954161969585121050, −8.461702348427371327175540546593, −7.78995882703487576802250501185, −6.86148830728735699344081253287, −5.66933075072515077638488772705, −5.22878621451135242859693604857, −3.83259936320929575497840574195, −2.88240640880334423766979253980, −1.84007323661342107435031624584,
1.09593256446608352525394974499, 2.56549309293035955227356864265, 3.53477729072207506457770332839, 4.31546429257068465726390181989, 5.27414225436957034825701011275, 6.39570300728915969287789123280, 7.46692139575242708796787737526, 8.156048795037560563526338378074, 9.340499684211932504041789203246, 9.748747873644460520208775182025