L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)19-s + (0.173 − 0.984i)20-s + (0.939 − 0.342i)22-s + (−0.5 − 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.939 − 0.342i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)19-s + (0.173 − 0.984i)20-s + (0.939 − 0.342i)22-s + (−0.5 − 0.866i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4332023122 - 0.5119508288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4332023122 - 0.5119508288i\) |
\(L(1)\) |
\(\approx\) |
\(0.9610184238 + 0.1410954777i\) |
\(L(1)\) |
\(\approx\) |
\(0.9610184238 + 0.1410954777i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.173 + 0.984i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−29.69746857800011108129553288898, −28.378046809170889315564684530303, −27.909036634600571781992608832390, −26.43695174987885711680228162105, −25.3524169740314835980912165036, −23.91094154040399070873903527607, −23.25336913761690283733942428228, −22.25023805190512440372657678053, −21.480203451193165815119689932689, −19.87823596905128171444944679648, −19.48098326039906436975445304769, −18.47103137024735441419602661468, −16.68183447163073616659639758099, −15.36648196394103525905893738937, −14.80802146879516786652329905699, −13.3266128078087492200885969908, −12.27319618132886316516128314018, −11.521352297475513434828530220213, −10.20662176543016258959828261238, −9.07894225085123568358913635024, −7.15479975400667601036246650569, −6.15042713510853149314937428580, −4.40588420411080755137566293988, −3.52926260187960309220975810618, −2.017122597865345573863561093463,
0.209941474266250669712946984092, 3.12478703579470729847771269494, 3.99972481283005711547567462324, 5.46431874829372495591315701585, 6.73628928780369318369490399535, 7.84160728158353674475092373033, 8.96059799142483669735999122832, 10.790706464354140638164299485679, 12.13998173991526431593326861758, 12.87952082311047550195330654673, 14.102683716613335650796260390606, 15.24925549592557297825715556976, 16.31125977792587148712068369038, 16.774029364155873458459809832597, 18.49749360263072615166979681334, 19.80423720601965483071958310902, 20.6153315330054134099970039315, 22.141621539024135196477241927295, 22.79736107120101108104269429803, 23.73541501560799822539094802338, 24.704735587114902500405048222098, 25.6492901935239943078261671901, 26.8485236753557029396426447477, 27.57527774413190202082959386770, 29.28718751048357307280333018054