L(s) = 1 | + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·21-s + (−0.5 − 0.866i)23-s − 25-s − 27-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·21-s + (−0.5 − 0.866i)23-s − 25-s − 27-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9730201566 + 0.2406863629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9730201566 + 0.2406863629i\) |
\(L(1)\) |
\(\approx\) |
\(1.115542137 + 0.1926393906i\) |
\(L(1)\) |
\(\approx\) |
\(1.115542137 + 0.1926393906i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 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
−33.68252336536203620806692948423, −31.9832973936763093747543366608, −30.90550810013843931871090231839, −30.06530802642846333267329106032, −29.280726852082216169272719725220, −27.507025392064643183606484573803, −26.30231923239819669484577191723, −25.530657732546820449500331370699, −23.94749273056359017369558586828, −23.409973257568655736407087037848, −21.687905082227729903282352851367, −20.48577630516190729637487005718, −19.09802827429505360552467942147, −18.31377542429120413444888674120, −17.15539069778570406554796692127, −15.12077324070295971124778728460, −14.22943483290959525382224468610, −13.1286616294813146035455421259, −11.51458378740287426414035550289, −10.31977773516113258815759993293, −8.27402548102020332203683418405, −7.40639648583334680534506335241, −5.92109750156327252711191868739, −3.575537049712194291785502663057, −1.96277225190173613044656552891,
2.33993580444931292873855472596, 4.44441703944746924986139708699, 5.32653210231909546308575715373, 7.92947442382310076582250579505, 8.87640419662040725682587795929, 10.18609695282364257068897489772, 11.713471581629377526371759732499, 13.196069879659542207295953049358, 14.641823126569829591916960772783, 15.69897273316911315954608072928, 16.80010213507133469175813663063, 18.23172873602840391860350065980, 19.91386934854839961278518131794, 20.863857142904898660331951440515, 21.5308760340493286533156075710, 23.19872760524660128053501741625, 24.55307985464701800341712220842, 25.49341386683074255505170454282, 26.83545300236801771562270749512, 27.92111292178228931259658935422, 28.56323226098773587679925198532, 30.44775188966919322294088010740, 31.59354152304034093698577513673, 32.198874478200424875395174707885, 33.535369857879609972592661680418