L(s) = 1 | + 1.89i·2-s + (−2.98 − 0.311i)3-s + 0.398·4-s + 8.25i·5-s + (0.592 − 5.66i)6-s + 3.60·7-s + 8.34i·8-s + (8.80 + 1.86i)9-s − 15.6·10-s + (−1.18 − 0.124i)12-s + 13.1·13-s + 6.84i·14-s + (2.57 − 24.6i)15-s − 14.2·16-s + 25.6i·17-s + (−3.53 + 16.7i)18-s + ⋯ |
L(s) = 1 | + 0.948i·2-s + (−0.994 − 0.103i)3-s + 0.0995·4-s + 1.65i·5-s + (0.0986 − 0.943i)6-s + 0.515·7-s + 1.04i·8-s + (0.978 + 0.206i)9-s − 1.56·10-s + (−0.0989 − 0.0103i)12-s + 1.00·13-s + 0.488i·14-s + (0.171 − 1.64i)15-s − 0.890·16-s + 1.50i·17-s + (−0.196 + 0.928i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0731102 + 1.40227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0731102 + 1.40227i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.98 + 0.311i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.89iT - 4T^{2} \) |
| 5 | \( 1 - 8.25iT - 25T^{2} \) |
| 7 | \( 1 - 3.60T + 49T^{2} \) |
| 13 | \( 1 - 13.1T + 169T^{2} \) |
| 17 | \( 1 - 25.6iT - 289T^{2} \) |
| 19 | \( 1 - 12.5T + 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 9.48iT - 841T^{2} \) |
| 31 | \( 1 + 5.08T + 961T^{2} \) |
| 37 | \( 1 + 34.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 27.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 43.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 107.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 69.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.0T + 9.40e3T^{2} \) |
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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
−11.34180030891588148576585829584, −10.83392520909760308518117880182, −10.26828130869401688566466024691, −8.474421273683446338029297676282, −7.56339429528171626325734094192, −6.62940295471663441784935166228, −6.26094251472231291609164363835, −5.21761122918855938607816785608, −3.64416542957247269723217275047, −1.94480993419065661402952168846,
0.76662869535222611198187164275, 1.60150975856027996411476577044, 3.64817410749758739240783256946, 4.81705893185234000399945979268, 5.51663598661810573293420350806, 6.85860696873900658279688325558, 8.060713492432782144432162606130, 9.332338757672133290949284011520, 9.865804184271321201583378095982, 11.25542750625243469257676578516