| L(s) = 1 | + (−0.212 − 1.98i)2-s + (−1.16 − 2.02i)3-s + (−3.90 + 0.843i)4-s + (1.55 − 2.68i)5-s + (−3.77 + 2.75i)6-s + (−6.89 − 1.21i)7-s + (2.50 + 7.59i)8-s + (1.77 − 3.06i)9-s + (−5.66 − 2.51i)10-s + (4.06 − 2.34i)11-s + (6.27 + 6.92i)12-s + 6.88·13-s + (−0.953 + 13.9i)14-s − 7.24·15-s + (14.5 − 6.59i)16-s + (14.7 − 8.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 − 0.994i)2-s + (−0.389 − 0.674i)3-s + (−0.977 + 0.210i)4-s + (0.310 − 0.537i)5-s + (−0.629 + 0.458i)6-s + (−0.984 − 0.173i)7-s + (0.313 + 0.949i)8-s + (0.196 − 0.341i)9-s + (−0.566 − 0.251i)10-s + (0.369 − 0.213i)11-s + (0.522 + 0.576i)12-s + 0.529·13-s + (−0.0681 + 0.997i)14-s − 0.482·15-s + (0.910 − 0.412i)16-s + (0.865 − 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.231956 - 0.836045i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231956 - 0.836045i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 + 1.98i)T \) |
| 7 | \( 1 + (6.89 + 1.21i)T \) |
| good | 3 | \( 1 + (1.16 + 2.02i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.55 + 2.68i)T + (-12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-4.06 + 2.34i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 6.88T + 169T^{2} \) |
| 17 | \( 1 + (-14.7 + 8.49i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.1 + 22.7i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (12.9 - 22.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 42.2iT - 841T^{2} \) |
| 31 | \( 1 + (-15.9 + 9.18i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (43.1 + 24.9i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 10.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-11.8 - 6.84i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (6.03 - 3.48i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-53.0 - 91.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-46.7 + 80.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-77.2 + 44.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 81.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (119. - 68.9i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-6.55 + 11.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 2.15T + 6.88e3T^{2} \) |
| 89 | \( 1 + (87.8 + 50.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 88.9iT - 9.40e3T^{2} \) |
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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
−13.97866665315745147921825663761, −13.11785855910450912497464473970, −12.34688998742148566613370651034, −11.33218580742548865660978289838, −9.811461513869953440316882557896, −8.991190177341185687625768995241, −7.13437336809418507513624277170, −5.50194995216065199489063961083, −3.46529700807003643905210860611, −1.04368625167428668651383189224,
3.89733521974626539448920660612, 5.63713392951164768847035310466, 6.63641991657068038611479540898, 8.218383586407558587058538983159, 9.874349860113360847093990292553, 10.26318769207452259710596298602, 12.23548529399825690319969336766, 13.57671604203089027018881634073, 14.55884087649315500310925175814, 15.74160477429361707402186586435
