L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − 14-s + 16-s + i·17-s + i·18-s + 19-s − 21-s + i·22-s + ⋯ |
L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − 14-s + 16-s + i·17-s + i·18-s + 19-s − 21-s + i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3695570296 - 0.6628391488i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3695570296 - 0.6628391488i\) |
\(L(1)\) |
\(\approx\) |
\(0.4097197214 - 0.6629404352i\) |
\(L(1)\) |
\(\approx\) |
\(0.4097197214 - 0.6629404352i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.76421617972623729726676073403, −31.56944923249218574601136857586, −31.24171123473786650650188874670, −28.95803419094287913273993086583, −27.92851310381020936779092254528, −27.01947603854789218692836183596, −25.969597850321002103977668614028, −25.11397479204308615333878166747, −23.7953321609730695738799987006, −22.54536526221187387989064245710, −21.76846758517232917750076188165, −20.53494960512183441830649917277, −18.740732273854421293077363214729, −17.73797521296180902044128825932, −16.26659549390168773048053462531, −15.63211386460687500551000774696, −14.64140163333971153728065132301, −13.29781052922602675923375034575, −11.589312047849249325603612442422, −9.86861047716615103129680768, −8.974308796279650592665140232656, −7.63028702906795903558473536279, −5.71608039558275707933346658337, −4.96461296713060060082698260418, −3.16650970393109837971195712624,
0.39184480300428952573422920640, 1.9703836689085405062464187484, 3.58367883989269684515199529165, 5.41552356075866429895562411336, 7.305271029617953847303244347751, 8.485565418463994628191552516707, 10.215960135529180046592771840994, 11.25948354792155985136902641183, 12.66404959536644353718705689311, 13.3949005881401251119789296790, 14.51577844851933707436307334913, 16.74675217499687804785617360013, 17.8970685259853354734479554077, 18.78907940802136336618600239613, 19.95255066224156165858986553104, 20.71456924587264684913341939995, 22.29874431025155860588918315258, 23.38778903570555983678039375267, 24.12635177655701527792065644199, 25.92642370030320721848675810338, 26.76311938321296310671869219267, 28.36721855947621313906891060043, 29.080336009712332549022597510405, 30.121622106617950563338806991614, 30.807790262876180284086190038120