| L(s) = 1 | − 4i·2-s − 11.2i·3-s − 16·4-s + (−42.1 + 36.7i)5-s − 44.8·6-s + 166. i·7-s + 64i·8-s + 117.·9-s + (147. + 168. i)10-s − 290.·11-s + 179. i·12-s + 151. i·13-s + 666.·14-s + (412. + 472. i)15-s + 256·16-s − 1.18e3i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.719i·3-s − 0.5·4-s + (−0.753 + 0.657i)5-s − 0.508·6-s + 1.28i·7-s + 0.353i·8-s + 0.481·9-s + (0.465 + 0.532i)10-s − 0.723·11-s + 0.359i·12-s + 0.248i·13-s + 0.909·14-s + (0.473 + 0.542i)15-s + 0.250·16-s − 0.996i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9868783240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9868783240\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 + (42.1 - 36.7i)T \) |
| 23 | \( 1 + 529iT \) |
| good | 3 | \( 1 + 11.2iT - 243T^{2} \) |
| 7 | \( 1 - 166. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 151. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 152.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.27e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 3.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.75e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.50e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 4.99e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.52e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 8.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.87e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.82e4iT - 8.58e9T^{2} \) |
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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
−11.22641557456945281167073094983, −10.12602480406220794314683899759, −9.031076687536568446762567049557, −7.948811399967378620807794484253, −7.08794728636605748675732166854, −5.79164050011547658192857926795, −4.43171910605687347525871629232, −2.93516164755068566608440547392, −2.08124232505233071652569095509, −0.34722896039569666865200692095,
1.04998439533586968025766766351, 3.65217401380082363275189744975, 4.34714446333104182479478332126, 5.28968799257293212068453272989, 6.85631241486513933161260280592, 7.77958087094992325696369639066, 8.535771480221812324992214160125, 9.910404595727466194212703612644, 10.43359936422327131625110975514, 11.63064990167618670042986884892
