L(s) = 1 | + (1.52 + 0.812i)3-s − 3.79·5-s + (2.59 + 0.525i)7-s + (1.67 + 2.48i)9-s + 4.51·11-s + (0.588 − 1.01i)13-s + (−5.81 − 3.08i)15-s + (−2.95 + 5.12i)17-s + (2.55 + 4.42i)19-s + (3.53 + 2.91i)21-s − 4.18·23-s + 9.43·25-s + (0.545 + 5.16i)27-s + (2.11 + 3.65i)29-s + (3.12 + 5.40i)31-s + ⋯ |
L(s) = 1 | + (0.883 + 0.469i)3-s − 1.69·5-s + (0.980 + 0.198i)7-s + (0.559 + 0.828i)9-s + 1.36·11-s + (0.163 − 0.282i)13-s + (−1.50 − 0.797i)15-s + (−0.717 + 1.24i)17-s + (0.586 + 1.01i)19-s + (0.772 + 0.635i)21-s − 0.871·23-s + 1.88·25-s + (0.105 + 0.994i)27-s + (0.392 + 0.679i)29-s + (0.560 + 0.971i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44615 + 0.817236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44615 + 0.817236i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.52 - 0.812i)T \) |
| 7 | \( 1 + (-2.59 - 0.525i)T \) |
good | 5 | \( 1 + 3.79T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + (-0.588 + 1.01i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.95 - 5.12i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 4.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + (-2.11 - 3.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.12 - 5.40i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.754 + 1.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.01 + 8.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.49 + 11.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.19 + 10.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.729 - 1.26i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.813 + 1.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.920 + 1.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.307 + 0.532i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.25 + 2.17i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 4.10i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.03587125850945836308237696023, −10.31109156609223744706163795488, −8.906706545203438787701864160394, −8.394638956315315346912866594217, −7.78971784332374308135131843233, −6.74672321916882861902709160050, −5.08304625758281439145941250066, −3.92631421732015883650932302318, −3.65778219321076395855021593230, −1.72821344745935968289087023886,
1.05148729461789996231270740699, 2.78565067331642433849305780461, 4.07882170991144289727239792062, 4.54565446913994212800013863175, 6.59341277911822438298682645000, 7.32792901821978356126772718479, 8.045242120812027369151352147581, 8.765980165894184633390603528111, 9.607278254355516265610037449612, 11.18519250542660508371364116128