L(s) = 1 | + 4.98·2-s − 6.26i·3-s + 16.8·4-s − 15.7i·5-s − 31.1i·6-s + 0.789i·7-s + 43.8·8-s − 12.2·9-s − 78.4i·10-s + 45.3i·11-s − 105. i·12-s + 46.7·13-s + 3.93i·14-s − 98.6·15-s + 84.1·16-s + ⋯ |
L(s) = 1 | + 1.76·2-s − 1.20i·3-s + 2.10·4-s − 1.40i·5-s − 2.12i·6-s + 0.0426i·7-s + 1.93·8-s − 0.452·9-s − 2.48i·10-s + 1.24i·11-s − 2.53i·12-s + 0.997·13-s + 0.0751i·14-s − 1.69·15-s + 1.31·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.272205116\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.272205116\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 4.98T + 8T^{2} \) |
| 3 | \( 1 + 6.26iT - 27T^{2} \) |
| 5 | \( 1 + 15.7iT - 125T^{2} \) |
| 7 | \( 1 - 0.789iT - 343T^{2} \) |
| 11 | \( 1 - 45.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 46.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 101. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 7.36iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 251. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 260. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 126. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 150.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 434. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 493. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 72.2iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 711.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3iT - 9.12e5T^{2} \) |
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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
−11.85778534529216986028602669792, −10.58626284151782150712580350408, −8.954954126132961850085602491541, −7.891338122682839357542505688047, −6.79118623708185038619141652119, −6.03919989943313974894556990546, −4.83747341570675182554331122599, −4.13253088518212597321825024083, −2.31709296793214027740872171486, −1.26280709648573248284289355574,
2.55475810481551180532941545873, 3.64761336082154479816490529361, 4.04769920145317820419438683074, 5.58461809481282927766587002741, 6.19650587579216354658422566531, 7.28473969773933001288852488212, 8.863208102867133343035503391654, 10.34871390085966874807276845544, 10.97979694700039700528758303656, 11.33665514474235702901249879348