L(s) = 1 | + (1 + 1.73i)5-s + (−0.5 + 0.866i)7-s + (1 − 1.73i)11-s + (−0.5 − 0.866i)13-s − 6·17-s − 5·19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + (4 − 6.92i)29-s + (−4 − 6.92i)31-s − 1.99·35-s − 5·37-s + (4 + 6.92i)41-s + (2 − 3.46i)43-s + (5 − 8.66i)47-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (−0.188 + 0.327i)7-s + (0.301 − 0.522i)11-s + (−0.138 − 0.240i)13-s − 1.45·17-s − 1.14·19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + (0.742 − 1.28i)29-s + (−0.718 − 1.24i)31-s − 0.338·35-s − 0.821·37-s + (0.624 + 1.08i)41-s + (0.304 − 0.528i)43-s + (0.729 − 1.26i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9161205837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9161205837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.649944092255145522756546529039, −8.046976466648692633894050132124, −6.90399314407430787810710999240, −6.33343729755726719893649220422, −5.87392053169782770705470233369, −4.59009484245182620070375389666, −3.89219501947325747785986371395, −2.58888551232916792624614062988, −2.19215591558151794541522193288, −0.28553088501887923982534574652,
1.38136835743757724794755140731, 2.20401594086502327641214219011, 3.54820953935455949684083733370, 4.45531722659863800064174109647, 5.05229215526127219463454538780, 6.04797544319859575536319676420, 6.82889127460352438549083489107, 7.44434627837536520937000682171, 8.634537837675451359002650181598, 9.009022716472264418779033159351