| L(s) = 1 | − 5.09i·2-s + 9i·3-s + 6.03·4-s + (54.3 + 12.9i)5-s + 45.8·6-s − 170. i·7-s − 193. i·8-s − 81·9-s + (66.0 − 277. i)10-s + 121·11-s + 54.3i·12-s + 364. i·13-s − 867.·14-s + (−116. + 489. i)15-s − 794.·16-s − 1.30e3i·17-s + ⋯ |
| L(s) = 1 | − 0.900i·2-s + 0.577i·3-s + 0.188·4-s + (0.972 + 0.231i)5-s + 0.520·6-s − 1.31i·7-s − 1.07i·8-s − 0.333·9-s + (0.208 − 0.876i)10-s + 0.301·11-s + 0.108i·12-s + 0.597i·13-s − 1.18·14-s + (−0.133 + 0.561i)15-s − 0.775·16-s − 1.09i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.554535609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554535609\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9iT \) |
| 5 | \( 1 + (-54.3 - 12.9i)T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 5.09iT - 32T^{2} \) |
| 7 | \( 1 + 170. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 364. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.30e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 619.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 922. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.28e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.73e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 6.94e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.73e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.43e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.94e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.59e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.43e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.65e4iT - 8.58e9T^{2} \) |
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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
−11.36507631991128373786691918563, −10.56968747292997289544468735655, −9.973702862039611603965121576788, −9.069255044436874455400700479513, −7.22374185950733724621651005780, −6.38562056618916788851875847556, −4.69531485860043012532468167129, −3.52814808800584066245689333493, −2.23449639226399440893065239525, −0.828195095109542465395994620679,
1.64025557458624815894456151442, 2.70004026706117395433036159418, 5.17596859610787839179637578844, 6.00387676588619751293081943239, 6.63799508263420374357161675827, 8.158382154513941139878171767413, 8.729696541776550185132139138628, 10.06139885172830453108774371663, 11.38491346506726689589871649706, 12.37526745604520319908500699897
