L(s) = 1 | + (−1.16 + 0.926i)2-s + (0.0461 − 0.202i)4-s + (2.75 − 1.32i)5-s + (−2.64 − 0.00862i)7-s + (−1.15 − 2.39i)8-s + (−1.97 + 4.09i)10-s + (4.38 − 3.49i)11-s + (−0.126 + 0.100i)13-s + (3.08 − 2.44i)14-s + (3.93 + 1.89i)16-s + (−0.351 − 1.53i)17-s − 5.64i·19-s + (−0.141 − 0.618i)20-s + (−1.85 + 8.12i)22-s + (−4.97 − 1.13i)23-s + ⋯ |
L(s) = 1 | + (−0.821 + 0.654i)2-s + (0.0230 − 0.101i)4-s + (1.23 − 0.594i)5-s + (−0.999 − 0.00325i)7-s + (−0.408 − 0.848i)8-s + (−0.624 + 1.29i)10-s + (1.32 − 1.05i)11-s + (−0.0350 + 0.0279i)13-s + (0.823 − 0.652i)14-s + (0.984 + 0.474i)16-s + (−0.0852 − 0.373i)17-s − 1.29i·19-s + (−0.0315 − 0.138i)20-s + (−0.395 + 1.73i)22-s + (−1.03 − 0.236i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.966813 - 0.0777087i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.966813 - 0.0777087i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.00862i)T \) |
good | 2 | \( 1 + (1.16 - 0.926i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.75 + 1.32i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 3.49i)T + (2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.126 - 0.100i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.351 + 1.53i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + 5.64iT - 19T^{2} \) |
| 23 | \( 1 + (4.97 + 1.13i)T + (20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-3.45 + 0.788i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 0.856iT - 31T^{2} \) |
| 37 | \( 1 + (-0.520 - 2.28i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 5.00i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-10.9 - 5.26i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-6.32 - 7.92i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (6.39 + 1.45i)T + (47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (8.14 + 3.92i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (0.616 - 0.140i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + (4.40 + 1.00i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.53 - 4.41i)T + (16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 + (-4.61 + 5.78i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.71 + 2.14i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 - 6.81iT - 97T^{2} \) |
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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
−10.84240914370154153112747748710, −9.631358085312949748062024821856, −9.260318068315510151123521791405, −8.706668666361967111202281675787, −7.39983920088199155009811471573, −6.24948885553440506732036544521, −6.03705374843359501291930524911, −4.27041392045129055513220996668, −2.86476016817683317249192028470, −0.869756168758315076086778784472,
1.56389214938347481107439079598, 2.54426053097837690695157566357, 4.02281896425827372007996347369, 5.86824720232625761090965621122, 6.26049990356900638611248589293, 7.51068859593684537772134324157, 8.970150426112242884047566574791, 9.567546641287978108866707325855, 10.09983305280705370232323861921, 10.71421420606255911947158208657