| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 35.4i·5-s + 93.3·7-s − 22.6i·8-s + 100.·10-s + 57.8i·11-s + 22.7·13-s + 264. i·14-s + 64.0·16-s − 359. i·17-s + 424.·19-s + 283. i·20-s − 163.·22-s − 217. i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.41i·5-s + 1.90·7-s − 0.353i·8-s + 1.00·10-s + 0.478i·11-s + 0.134·13-s + 1.34i·14-s + 0.250·16-s − 1.24i·17-s + 1.17·19-s + 0.709i·20-s − 0.338·22-s − 0.411i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.71233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71233\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 35.4iT - 625T^{2} \) |
| 7 | \( 1 - 93.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 57.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 22.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 359. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 424.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 217. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 152. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 942.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.33e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.76e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.31e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.08e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.66e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.29e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.65e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.26e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.97e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.52e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.02e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.53e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.50e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.32e3T + 8.85e7T^{2} \) |
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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
−14.58882306757505043464781354246, −13.70807372021050023741951767423, −12.37458714102485606786057499644, −11.32958028718591067116873079101, −9.450815670943940071319070473780, −8.406715647716478048249486298447, −7.47021063802836891929165565545, −5.25228926457999451062123976604, −4.66504091588126538163329285536, −1.26721762148500223949189950448,
1.83478905229235413994641605802, 3.62626130915886602052590877592, 5.45570448561220037766443098899, 7.36932935266496174644784991867, 8.566307815326426324135323642326, 10.38691995029766219840567698987, 11.07596304458746757732372054211, 11.89348098963337437727930009954, 13.71592863094937237798549765784, 14.44916877926601649076243812924
