L(s) = 1 | + (0.173 + 0.984i)3-s + (0.766 − 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + (0.939 + 0.342i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)3-s + (0.766 − 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (0.866 − 0.5i)23-s + (−0.5 − 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + (0.939 + 0.342i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.779115677 - 0.08499003105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779115677 - 0.08499003105i\) |
\(L(1)\) |
\(\approx\) |
\(1.226744385 + 0.1331058757i\) |
\(L(1)\) |
\(\approx\) |
\(1.226744385 + 0.1331058757i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.342 - 0.939i)T \) |
| 17 | \( 1 + (0.342 + 0.939i)T \) |
| 19 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.642 + 0.766i)T \) |
| 61 | \( 1 + (0.342 - 0.939i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.642 - 0.766i)T \) |
| 83 | \( 1 + (0.939 - 0.342i)T \) |
| 89 | \( 1 + (0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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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
−20.74802258470469949682091480866, −19.809712511733256138896077434028, −19.01436081621824426429732706221, −18.52825948754790686434242628579, −17.72306419650976121247995109512, −17.16685935408584442732178125994, −16.2631029304095145211796893306, −15.05818684973208621152059319733, −14.642760368497430680130272920912, −13.89770030302581731350336543634, −12.965343137006368890700565708259, −12.36277335078182018115973556127, −11.41045939390509175818969351148, −11.206276489771854468948330923048, −9.54042570361548709217938273671, −9.07357301318747928628420646610, −8.19052734735405360593601962522, −7.35201900479523535677076845021, −6.727261652057424897466489666732, −5.804016772682330970267856827539, −4.89930802268172335282936132326, −3.938340757214209670398655304188, −2.64159638108827043268911373843, −1.96028399671769756016706145999, −1.140495004118746435143160419284,
0.73873353332924094044853051794, 1.99364603228872766448595424934, 3.28087896601936819642757672758, 3.82928529585136385959906302170, 4.70476909532767743062092287266, 5.576275074534356499603400220081, 6.348820603674129176074610286901, 7.62735695024932744842609219122, 8.42788297864568766312842943715, 8.869945549997980876192465676882, 10.061049772694244502915565750059, 10.81534540094155779188878118743, 11.01320245912347855090209199499, 12.19061729780817755371338014178, 13.21222178739196321908648955585, 13.97241267306631553681908741231, 14.79721704935924253614402680085, 15.143708484769033696308189495505, 16.22002179705356454671538962069, 16.98765082038996879845215718949, 17.27170620266373074181482502149, 18.41166398114321706227991912632, 19.365399007400686559381093589095, 19.97561967449147709531825550971, 20.799711519109437146540928926776