L(s) = 1 | + (0.797 + 5.60i)2-s + 16.3·3-s + (−30.7 + 8.93i)4-s − 79.2·5-s + (13.0 + 91.7i)6-s + 58.5i·7-s + (−74.5 − 164. i)8-s + 25.6·9-s + (−63.2 − 443. i)10-s − 540. i·11-s + (−503. + 146. i)12-s + 72.5i·13-s + (−327. + 46.6i)14-s − 1.29e3·15-s + (864. − 548. i)16-s − 1.33e3·17-s + ⋯ |
L(s) = 1 | + (0.140 + 0.990i)2-s + 1.05·3-s + (−0.960 + 0.279i)4-s − 1.41·5-s + (0.148 + 1.04i)6-s + 0.451i·7-s + (−0.411 − 0.911i)8-s + 0.105·9-s + (−0.199 − 1.40i)10-s − 1.34i·11-s + (−1.00 + 0.293i)12-s + 0.119i·13-s + (−0.446 + 0.0636i)14-s − 1.49·15-s + (0.844 − 0.536i)16-s − 1.11·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0562328 - 0.0896001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0562328 - 0.0896001i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.797 - 5.60i)T \) |
| 19 | \( 1 + (1.05e3 - 1.16e3i)T \) |
good | 3 | \( 1 - 16.3T + 243T^{2} \) |
| 5 | \( 1 + 79.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 58.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 540. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 72.5iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.33e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 620. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 604. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.48e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.05e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.18e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.01e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.10e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.12e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.78e5iT - 8.58e9T^{2} \) |
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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
−14.45659993937907477853594601578, −13.54386841654981628819428355089, −12.33396340104990999455398732795, −11.05264696943345214453828909087, −9.018116035566872079360880569177, −8.446961735194228280774773139401, −7.61560168869448969974350462919, −6.08850448908247575394779901492, −4.26311178746963815183026953144, −3.16822475105627199844437166373,
0.03727464090700783001291290886, 2.24566649459319252353988107744, 3.69021717167392742982111954519, 4.57860196781859299525791487890, 7.25761493804113373755987765383, 8.363573810427507690446298511074, 9.321677012996508499407986747001, 10.70195091273842811251953357849, 11.69605801495828896110244803771, 12.76009059386672326989749393507