| L(s) = 1 | + (−2.22 + 0.873i)2-s + (−0.408 − 1.68i)3-s + (2.72 − 2.52i)4-s + (−2.05 − 0.989i)5-s + (2.37 + 3.38i)6-s + (2.60 + 0.441i)7-s + (−1.77 + 3.68i)8-s + (−2.66 + 1.37i)9-s + (5.43 + 0.407i)10-s + (4.06 + 3.24i)11-s + (−5.35 − 3.54i)12-s + (−0.574 + 0.225i)13-s + (−6.18 + 1.29i)14-s + (−0.826 + 3.86i)15-s + (0.175 − 2.34i)16-s + (1.39 + 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−1.57 + 0.617i)2-s + (−0.235 − 0.971i)3-s + (1.36 − 1.26i)4-s + (−0.919 − 0.442i)5-s + (0.970 + 1.38i)6-s + (0.985 + 0.166i)7-s + (−0.627 + 1.30i)8-s + (−0.888 + 0.458i)9-s + (1.71 + 0.128i)10-s + (1.22 + 0.977i)11-s + (−1.54 − 1.02i)12-s + (−0.159 + 0.0625i)13-s + (−1.65 + 0.346i)14-s + (−0.213 + 0.997i)15-s + (0.0439 − 0.585i)16-s + (0.339 + 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.544584 - 0.179182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.544584 - 0.179182i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.408 + 1.68i)T \) |
| 7 | \( 1 + (-2.60 - 0.441i)T \) |
| good | 2 | \( 1 + (2.22 - 0.873i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (2.05 + 0.989i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-4.06 - 3.24i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.574 - 0.225i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.39 - 1.29i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.54 + 2.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.610 - 0.139i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (2.22 + 7.20i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-4.06 + 2.34i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.42 - 1.67i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-6.01 + 4.10i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-4.94 - 3.36i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (0.914 + 2.32i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 5.51i)T + (-43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-10.4 - 7.09i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (0.0705 - 0.0760i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (6.07 + 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 0.604i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.02 + 6.79i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (6.67 - 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.52 + 14.0i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (2.61 - 6.67i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (10.2 - 5.93i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.12569325970508770299675943046, −9.833799370976699888785972784566, −8.937489027304230006596434935902, −8.125918186903842152827414798191, −7.56198356783350071439903703276, −6.88648426306223331404409578244, −5.73398747285149996801447852177, −4.34295165652699077812237467058, −1.97325744168771686576341300673, −0.836837234977441729322687485279,
1.09366758510614277603876490118, 3.09637182737735302368039538088, 3.96803463555115562949848653432, 5.45542604839149302544507176866, 7.01750897172711504354532705029, 7.920504405671503790110913860632, 8.700708056959835115366056494901, 9.409385531441077348324054039881, 10.41907750903698442833010862384, 11.05909672347555642133031071882
