| L(s) = 1 | + (0.658 + 1.29i)2-s + (−0.464 − 1.66i)3-s + (−0.0598 + 0.0824i)4-s + (−0.504 − 3.18i)5-s + (1.85 − 1.69i)6-s + (1.19 + 1.39i)7-s + (2.71 + 0.430i)8-s + (−2.56 + 1.54i)9-s + (3.78 − 2.74i)10-s + (−2.14 + 3.49i)11-s + (0.165 + 0.0616i)12-s + (−0.113 − 1.44i)13-s + (−1.01 + 2.45i)14-s + (−5.08 + 2.32i)15-s + (1.29 + 3.98i)16-s + (1.38 + 5.76i)17-s + ⋯ |
| L(s) = 1 | + (0.465 + 0.913i)2-s + (−0.267 − 0.963i)3-s + (−0.0299 + 0.0412i)4-s + (−0.225 − 1.42i)5-s + (0.755 − 0.693i)6-s + (0.450 + 0.527i)7-s + (0.960 + 0.152i)8-s + (−0.856 + 0.516i)9-s + (1.19 − 0.869i)10-s + (−0.646 + 1.05i)11-s + (0.0477 + 0.0178i)12-s + (−0.0314 − 0.399i)13-s + (−0.272 + 0.657i)14-s + (−1.31 + 0.599i)15-s + (0.323 + 0.997i)16-s + (0.335 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27706 - 0.142932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27706 - 0.142932i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.464 + 1.66i)T \) |
| 41 | \( 1 + (5.55 - 3.19i)T \) |
| good | 2 | \( 1 + (-0.658 - 1.29i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.504 + 3.18i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-1.19 - 1.39i)T + (-1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (2.14 - 3.49i)T + (-4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (0.113 + 1.44i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 5.76i)T + (-15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.256 + 3.25i)T + (-18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (-0.397 + 1.22i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.529 + 2.20i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (-3.77 - 5.20i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (7.03 + 5.11i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (7.75 - 3.95i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-0.836 - 0.714i)T + (7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (10.5 + 2.52i)T + (47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (-6.75 - 2.19i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.65 + 7.17i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-7.53 - 12.2i)T + (-30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (-5.32 - 3.26i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (8.25 + 8.25i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.314 + 0.758i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + (-5.17 + 4.41i)T + (13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (7.66 - 4.69i)T + (44.0 - 86.4i)T^{2} \) |
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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
−13.15855369036823122303838510892, −12.72623673282173132939939274096, −11.74693635198569081435182608194, −10.34175543138344382961249513494, −8.523628108889830775061016750067, −7.943019840848875778350480858422, −6.71192536089144739552020615161, −5.41080443292672201413670543885, −4.81188714777554925267284258415, −1.71903004181172820064466062576,
2.89433171003485510618164590754, 3.71064894280466461855876922834, 5.14809761989933504220077608478, 6.83168609648449056485642640359, 8.062201386842500087052683371892, 9.902409781525816085955595075937, 10.65296802474833703779587725080, 11.30218087605800570384532994237, 11.93312554892178573475448423275, 13.76955005378224706005055682292
