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
−33.535369857879609972592661680418, −32.198874478200424875395174707885, −31.59354152304034093698577513673, −30.44775188966919322294088010740, −28.56323226098773587679925198532, −27.92111292178228931259658935422, −26.83545300236801771562270749512, −25.49341386683074255505170454282, −24.55307985464701800341712220842, −23.19872760524660128053501741625, −21.5308760340493286533156075710, −20.863857142904898660331951440515, −19.91386934854839961278518131794, −18.23172873602840391860350065980, −16.80010213507133469175813663063, −15.69897273316911315954608072928, −14.641823126569829591916960772783, −13.196069879659542207295953049358, −11.713471581629377526371759732499, −10.18609695282364257068897489772, −8.87640419662040725682587795929, −7.92947442382310076582250579505, −5.32653210231909546308575715373, −4.44441703944746924986139708699, −2.33993580444931292873855472596,
1.96277225190173613044656552891, 3.575537049712194291785502663057, 5.92109750156327252711191868739, 7.40639648583334680534506335241, 8.27402548102020332203683418405, 10.31977773516113258815759993293, 11.51458378740287426414035550289, 13.1286616294813146035455421259, 14.22943483290959525382224468610, 15.12077324070295971124778728460, 17.15539069778570406554796692127, 18.31377542429120413444888674120, 19.09802827429505360552467942147, 20.48577630516190729637487005718, 21.687905082227729903282352851367, 23.409973257568655736407087037848, 23.94749273056359017369558586828, 25.530657732546820449500331370699, 26.30231923239819669484577191723, 27.507025392064643183606484573803, 29.280726852082216169272719725220, 30.06530802642846333267329106032, 30.90550810013843931871090231839, 31.9832973936763093747543366608, 33.68252336536203620806692948423