L(s) = 1 | − 1.03i·2-s + (−0.423 − 2.96i)3-s + 2.93·4-s + 2.23i·5-s + (−3.07 + 0.438i)6-s − 13.7·7-s − 7.16i·8-s + (−8.64 + 2.51i)9-s + 2.31·10-s − 17.1i·11-s + (−1.24 − 8.70i)12-s − 3.60·13-s + 14.2i·14-s + (6.64 − 0.947i)15-s + 4.31·16-s − 6.37i·17-s + ⋯ |
L(s) = 1 | − 0.516i·2-s + (−0.141 − 0.989i)3-s + 0.732·4-s + 0.447i·5-s + (−0.511 + 0.0730i)6-s − 1.96·7-s − 0.895i·8-s + (−0.960 + 0.279i)9-s + 0.231·10-s − 1.56i·11-s + (−0.103 − 0.725i)12-s − 0.277·13-s + 1.01i·14-s + (0.442 − 0.0631i)15-s + 0.269·16-s − 0.375i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0723930 - 1.01995i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0723930 - 1.01995i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.423 + 2.96i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 13 | \( 1 + 3.60T \) |
good | 2 | \( 1 + 1.03iT - 4T^{2} \) |
| 7 | \( 1 + 13.7T + 49T^{2} \) |
| 11 | \( 1 + 17.1iT - 121T^{2} \) |
| 17 | \( 1 + 6.37iT - 289T^{2} \) |
| 19 | \( 1 + 0.865T + 361T^{2} \) |
| 23 | \( 1 + 20.1iT - 529T^{2} \) |
| 29 | \( 1 - 12.0iT - 841T^{2} \) |
| 31 | \( 1 - 37.3T + 961T^{2} \) |
| 37 | \( 1 + 29.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 72.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 42.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 53.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 54.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.47T + 3.72e3T^{2} \) |
| 67 | \( 1 + 47.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 122. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.4T + 9.40e3T^{2} \) |
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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
−11.90461282714844852256668379191, −10.95148092389721173197476322580, −10.10080552228359306624179701710, −8.852579909597418205783223080926, −7.42077402632336090430084017135, −6.45182292608019808822149070748, −6.04794142317405036136874279070, −3.30027673304354134846516395283, −2.68406318001515219515689370665, −0.55495937800121098795933395726,
2.67524352492213851747345677767, 4.07712900185835919526460291194, 5.50537672801864496436564170486, 6.45628370746059892806364614928, 7.44801893639251403220717970477, 8.985693458494357812269733193021, 9.839871079475963986002267444764, 10.45783636879617172175469161842, 11.94698877333612992335266835382, 12.51004242351481875183183120434