| L(s) = 1 | + (3.53 − 7.17i)2-s + (−38.9 − 50.7i)4-s − 55.9·5-s − 99.9i·7-s + (−502. + 99.7i)8-s + (−197. + 401. i)10-s + 2.39e3i·11-s + 3.61e3·13-s + (−717. − 353. i)14-s + (−1.06e3 + 3.95e3i)16-s + 927.·17-s + 773. i·19-s + (2.17e3 + 2.83e3i)20-s + (1.72e4 + 8.49e3i)22-s − 1.22e4i·23-s + ⋯ |
| L(s) = 1 | + (0.442 − 0.896i)2-s + (−0.608 − 0.793i)4-s − 0.447·5-s − 0.291i·7-s + (−0.980 + 0.194i)8-s + (−0.197 + 0.401i)10-s + 1.80i·11-s + 1.64·13-s + (−0.261 − 0.128i)14-s + (−0.259 + 0.965i)16-s + 0.188·17-s + 0.112i·19-s + (0.272 + 0.354i)20-s + (1.61 + 0.797i)22-s − 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.180675498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180675498\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.53 + 7.17i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 55.9T \) |
| good | 7 | \( 1 + 99.9iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.39e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.61e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 927.T + 2.41e7T^{2} \) |
| 19 | \( 1 - 773. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.22e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.90e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 9.56e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 8.86e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.37e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.40e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.05e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 8.09e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 5.90e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.16e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.34e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.68e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 7.95e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.70e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.49e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.85e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.64e5T + 8.32e11T^{2} \) |
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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
−11.40487327342795312177343251227, −10.61054538749475510216833310427, −9.656162747367754586151688530265, −8.580774873533311068637123786633, −7.23022900267379154475851808592, −5.91714292428032022079870920865, −4.51211901571304702166445777401, −3.75994627025129677765157323189, −2.20674016812047167919679820906, −0.912055721343948812977818758482,
0.75376034933173012287648083826, 3.17212543952055801347880231894, 4.00237249778020830764447951042, 5.64667159629494554344736248219, 6.18043389263150141407449775467, 7.65987127986392388980576900549, 8.458955298793580799253046372294, 9.265504177117842888371930609198, 11.08284871011307819191982140957, 11.62320499654665607073056278178
