L(s) = 1 | + (2.82 + 2.83i)2-s + (−0.0429 + 15.9i)4-s + 11.7·5-s − 58.3i·7-s + (−45.4 + 45.0i)8-s + (33.2 + 33.3i)10-s − 100. i·11-s − 170.·13-s + (165. − 164. i)14-s + (−255. − 1.37i)16-s − 398.·17-s − 404. i·19-s + (−0.506 + 188. i)20-s + (284. − 283. i)22-s + 336. i·23-s + ⋯ |
L(s) = 1 | + (0.706 + 0.708i)2-s + (−0.00268 + 0.999i)4-s + 0.471·5-s − 1.19i·7-s + (−0.709 + 0.704i)8-s + (0.332 + 0.333i)10-s − 0.829i·11-s − 1.00·13-s + (0.843 − 0.841i)14-s + (−0.999 − 0.00536i)16-s − 1.37·17-s − 1.12i·19-s + (−0.00126 + 0.471i)20-s + (0.587 − 0.586i)22-s + 0.635i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00268 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00268 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.123386471\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123386471\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 - 2.83i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 11.7T + 625T^{2} \) |
| 7 | \( 1 + 58.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 100. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 170.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 398.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 404. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 336. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 655.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 635. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.59e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.46e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.23e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.29e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.15e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.92e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.56e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.63e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.49e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.22e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.15e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 910.T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.76e4T + 8.85e7T^{2} \) |
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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
−10.96634905518558002187250010577, −9.703737676665155225710746938621, −8.734933790064241688512631779759, −7.52345769342103227870692116193, −6.89594480271707817216528277722, −5.80946303943961245852717957883, −4.71969096780956633421060834522, −3.75180197520940623210746638103, −2.35747476584839266914661880332, −0.23407322963278600661215194163,
1.90539997584294328743501911763, 2.50730432211703775738333266599, 4.11832125106079964664230729488, 5.19562889172416274031927248587, 5.99937112280177813143699566826, 7.12883132516000005030531034753, 8.729792532078679704330209902079, 9.550042031501181901025545348080, 10.30331032718226903214153882239, 11.38009328028221395176285534502