L(s) = 1 | + (0.451 + 0.782i)2-s + (1.52 + 0.820i)3-s + (0.591 − 1.02i)4-s − 1.39·5-s + (0.0470 + 1.56i)6-s + (−0.0330 − 0.0190i)7-s + 2.87·8-s + (1.65 + 2.50i)9-s + (−0.631 − 1.09i)10-s + (−1.29 + 2.24i)11-s + (1.74 − 1.07i)12-s + (2.41 − 1.39i)13-s − 0.0345i·14-s + (−2.13 − 1.14i)15-s + (0.115 + 0.199i)16-s + (−3.52 + 2.03i)17-s + ⋯ |
L(s) = 1 | + (0.319 + 0.553i)2-s + (0.880 + 0.473i)3-s + (0.295 − 0.512i)4-s − 0.625·5-s + (0.0192 + 0.638i)6-s + (−0.0125 − 0.00721i)7-s + 1.01·8-s + (0.551 + 0.834i)9-s + (−0.199 − 0.345i)10-s + (−0.390 + 0.676i)11-s + (0.503 − 0.311i)12-s + (0.670 − 0.387i)13-s − 0.00922i·14-s + (−0.550 − 0.296i)15-s + (0.0288 + 0.0498i)16-s + (−0.854 + 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66056 + 0.662828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66056 + 0.662828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.52 - 0.820i)T \) |
| 67 | \( 1 + (7.49 - 3.28i)T \) |
good | 2 | \( 1 + (-0.451 - 0.782i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + (0.0330 + 0.0190i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.29 - 2.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.41 + 1.39i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.52 - 2.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.93 + 6.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.808 + 0.466i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.19 + 4.15i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.33 - 1.35i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.92 - 6.80i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.99 + 5.18i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 5.14iT - 43T^{2} \) |
| 47 | \( 1 + (2.55 + 1.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 - 0.806iT - 59T^{2} \) |
| 61 | \( 1 + (-0.740 + 0.427i)T + (30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.83 - 2.21i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.27 - 3.93i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.0 - 6.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 6.28i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.8iT - 89T^{2} \) |
| 97 | \( 1 + (-1.73 + 1.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.08004126974729189715153208304, −11.33060949172884889487979363903, −10.63677662722250569559759306538, −9.583687041766407898490386634035, −8.412162808742963910040131622786, −7.51721731154708028307641267768, −6.45804092037287990360776614171, −4.94427880047620592556036198523, −4.03684210993074003694653689534, −2.27672946008844617423201767639,
1.99122546934302984027425971324, 3.38426360553811283502504202356, 4.17152311286245049581552750397, 6.26230911971081906382990408393, 7.53748176029953522469843356500, 8.155977060192509256012081370227, 9.185849754857654103032575325867, 10.70406724512631947330801520229, 11.48468512860308332755086862491, 12.45223578321729963217630358577