L(s) = 1 | + (−1.61 + 1.28i)2-s + (0.505 − 2.21i)4-s + (−3.93 + 1.89i)5-s + (−2.02 − 1.70i)7-s + (0.245 + 0.509i)8-s + (3.91 − 8.12i)10-s + (3.17 − 2.53i)11-s + (−1.07 + 0.858i)13-s + (5.46 + 0.147i)14-s + (3.04 + 1.46i)16-s + (0.000551 + 0.00241i)17-s + 4.84i·19-s + (2.20 + 9.67i)20-s + (−1.87 + 8.19i)22-s + (1.08 + 0.246i)23-s + ⋯ |
L(s) = 1 | + (−1.14 + 0.911i)2-s + (0.252 − 1.10i)4-s + (−1.75 + 0.846i)5-s + (−0.764 − 0.644i)7-s + (0.0867 + 0.180i)8-s + (1.23 − 2.56i)10-s + (0.958 − 0.764i)11-s + (−0.298 + 0.238i)13-s + (1.46 + 0.0395i)14-s + (0.760 + 0.366i)16-s + (0.000133 + 0.000585i)17-s + 1.11i·19-s + (0.493 + 2.16i)20-s + (−0.398 + 1.74i)22-s + (0.225 + 0.0514i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.386616 + 0.0696096i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.386616 + 0.0696096i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.02 + 1.70i)T \) |
good | 2 | \( 1 + (1.61 - 1.28i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (3.93 - 1.89i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-3.17 + 2.53i)T + (2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.07 - 0.858i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.000551 - 0.00241i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 4.84iT - 19T^{2} \) |
| 23 | \( 1 + (-1.08 - 0.246i)T + (20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 0.361i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 + (2.10 + 9.23i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-7.61 + 3.66i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.421 + 0.203i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.51 + 1.89i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-6.81 - 1.55i)T + (47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (7.40 + 3.56i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-14.5 + 3.32i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + (4.38 + 1.00i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.34 + 3.46i)T + (16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + (-10.4 + 13.1i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.92 + 4.92i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + 2.91iT - 97T^{2} \) |
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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
−10.90841136114495915576610935725, −10.21929935892281375904168687456, −9.115840409289322727384154248703, −8.294747358872859040879486914163, −7.42613203121626997500736031399, −6.93649820100708836130380522732, −6.08472937719444590930341308946, −4.00879945880225176448636943232, −3.40407582603124805556138430555, −0.52418856849910823593981495863,
0.906002876707634631921142089349, 2.73137420229506195375827259183, 3.90879757430693220571906626005, 5.03903627762973987545143870082, 6.79665159633836677230456630953, 7.76849576480956743010234264296, 8.661975349508480524768182668173, 9.189591985702273105835910941807, 10.00992839246485992636361057230, 11.30771423817201096866913061783