L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s − 11-s + (0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (0.5 − 0.866i)20-s + (−0.939 − 0.342i)22-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s − 11-s + (0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (0.5 − 0.866i)20-s + (−0.939 − 0.342i)22-s + (−0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0151 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0151 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.590665846 - 1.566743402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590665846 - 1.566743402i\) |
\(L(1)\) |
\(\approx\) |
\(1.511735535 - 0.3008075209i\) |
\(L(1)\) |
\(\approx\) |
\(1.511735535 - 0.3008075209i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.766 - 0.642i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (0.766 - 0.642i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−27.85118601530501709909499632401, −26.19411029882850238419162857548, −25.76515972046651842247734294024, −24.362303551999703874295509521175, −23.52718142851485205244293355546, −22.62565257833313675841486459955, −21.72933083398686311347504769292, −21.16644876605773649823859943385, −19.64603105204991703633595507587, −18.969888053383699372534985096990, −18.074512253038656053394349838932, −16.24398416257586021491261923442, −15.44756102742412785246980264464, −14.63353080535064781246282178268, −13.55257352816512760301877680443, −12.53805826126639877630065466488, −11.50372498742353021017470638710, −10.615524972598995646625371690794, −9.5067027451109790564763977433, −7.76877388922571674291080661617, −6.47660173016732572742681649550, −5.728648297627940015960745178842, −4.18137113923975602762273663820, −2.99947107096344543144459408625, −2.03736271400193113384870584556,
0.552343279974136500987628507607, 2.60709869890968498551135469305, 3.95779606487175981495594296725, 4.957042576711063105289416173189, 6.01951024413931949690814765632, 7.4577671690148032116400089014, 8.20946580884552227038361790649, 9.87071228603845077259035672593, 11.08464826476514026019168754351, 12.36482335300805733698757217945, 13.15754801238342509535586718087, 13.809506884608896069497934897867, 15.33122144538432504662943025402, 16.08968451649959509855847400110, 16.83477392177625045266014047698, 18.00768900278833950105250476740, 19.735531866228566193226437263432, 20.46389600461842118066123421381, 21.092076464638514502248841470393, 22.57592064607762884099136324064, 23.15186708070689722627319369647, 24.12286596641480630022996383737, 24.8813078126177107574054486417, 25.93854716154996180817493449620, 26.833653959212270854545971622914