| L(s) = 1 | + (2.54 + 0.190i)2-s + (−0.0398 + 1.73i)3-s + (4.44 + 0.669i)4-s + (−2.54 − 2.36i)5-s + (−0.430 + 4.39i)6-s + (−1.65 − 2.06i)7-s + (6.19 + 1.41i)8-s + (−2.99 − 0.138i)9-s + (−6.02 − 6.49i)10-s + (−1.15 + 1.69i)11-s + (−1.33 + 7.66i)12-s + (1.64 + 3.42i)13-s + (−3.82 − 5.55i)14-s + (4.19 − 4.32i)15-s + (6.88 + 2.12i)16-s + (−0.225 + 0.573i)17-s + ⋯ |
| L(s) = 1 | + (1.79 + 0.134i)2-s + (−0.0230 + 0.999i)3-s + (2.22 + 0.334i)4-s + (−1.14 − 1.05i)5-s + (−0.175 + 1.79i)6-s + (−0.627 − 0.778i)7-s + (2.18 + 0.499i)8-s + (−0.998 − 0.0460i)9-s + (−1.90 − 2.05i)10-s + (−0.349 + 0.512i)11-s + (−0.385 + 2.21i)12-s + (0.457 + 0.949i)13-s + (−1.02 − 1.48i)14-s + (1.08 − 1.11i)15-s + (1.72 + 0.530i)16-s + (−0.0545 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.21350 + 0.589272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21350 + 0.589272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0398 - 1.73i)T \) |
| 7 | \( 1 + (1.65 + 2.06i)T \) |
| good | 2 | \( 1 + (-2.54 - 0.190i)T + (1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (2.54 + 2.36i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (1.15 - 1.69i)T + (-4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.64 - 3.42i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (0.225 - 0.573i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.476 - 0.275i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.58 + 1.80i)T + (16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.298 + 0.238i)T + (6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-7.17 + 4.14i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.59 - 0.693i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (0.952 - 4.17i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.956 + 4.19i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (0.0170 - 0.227i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (1.32 - 8.79i)T + (-50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (6.94 - 6.44i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.559 + 3.71i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (3.03 + 5.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.71 + 6.94i)T + (15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.106 - 0.00794i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-5.39 + 9.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.462 - 0.222i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-4.22 + 2.87i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + 15.0iT - 97T^{2} \) |
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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
−13.21599152714070225890009159276, −12.23505415990323355049690456180, −11.52931786942716342206726935731, −10.48671734923902785596317463833, −8.995095854029648072055292843436, −7.58370276957879461040066852444, −6.28307382480258066012836102199, −4.75534830116571300737359667369, −4.33890727654543701617502982335, −3.29942508362814522101063133733,
2.83519768436180663310998809313, 3.35830716204611432219639811387, 5.32330859104949912995472076859, 6.36789133190789919087282890220, 7.15129324659344262094116775236, 8.328880694508767577231330106952, 10.67321544828130099955399699414, 11.45612632431273815643650749562, 12.16664759777698088704111402729, 12.97963476889738254105799731091
