| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 5.18i·5-s − 12.5i·7-s − 2.82·8-s + 7.33i·10-s − 12.5i·11-s − 12.3·13-s + 17.7i·14-s + 4.00·16-s + 19.8i·17-s + 14.6i·19-s − 10.3i·20-s + 17.7i·22-s + (−22.8 + 2.14i)23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 1.03i·5-s − 1.78i·7-s − 0.353·8-s + 0.733i·10-s − 1.13i·11-s − 0.947·13-s + 1.26i·14-s + 0.250·16-s + 1.16i·17-s + 0.771i·19-s − 0.518i·20-s + 0.804i·22-s + (−0.995 + 0.0933i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0321546 - 0.687153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0321546 - 0.687153i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (22.8 - 2.14i)T \) |
| good | 5 | \( 1 + 5.18iT - 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 + 12.5iT - 121T^{2} \) |
| 13 | \( 1 + 12.3T + 169T^{2} \) |
| 17 | \( 1 - 19.8iT - 289T^{2} \) |
| 19 | \( 1 - 14.6iT - 361T^{2} \) |
| 29 | \( 1 + 6.85T + 841T^{2} \) |
| 31 | \( 1 - 24.7T + 961T^{2} \) |
| 37 | \( 1 - 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 3.48T + 1.68e3T^{2} \) |
| 43 | \( 1 + 45.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 17.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 50.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 6.44iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 61.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 121.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 114. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.2iT - 9.40e3T^{2} \) |
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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
−10.28133276158201603287333825816, −9.899508930749719318462872653657, −8.498658858085451814360312131890, −8.061685692082798034670352666550, −7.00652483887640100784806276600, −5.89467686215484795556484982730, −4.55174258601989727162233301131, −3.53199464702433724779542817022, −1.50282172703171653811297170491, −0.36913229992634862268326579721,
2.28380535096836880007948901300, 2.75920002082632551241346787088, 4.78423970939585097613622841944, 5.93028965588686715464330334772, 6.95818230246760581585290673696, 7.66830822356057142676490989766, 8.935201635156293729257486580373, 9.572251817236128839524032701255, 10.34827673801383164193703853211, 11.57142713431670752129938271769
