L(s) = 1 | + (−0.525 − 0.850i)2-s + (−0.447 + 0.894i)4-s + (−0.873 + 0.486i)5-s + (0.995 − 0.0896i)8-s + (0.873 + 0.486i)10-s + (0.280 − 0.959i)11-s + (−0.473 + 0.880i)13-s + (−0.599 − 0.800i)16-s + (0.998 + 0.0598i)17-s + (−0.913 + 0.406i)19-s + (−0.0448 − 0.998i)20-s + (−0.963 + 0.266i)22-s + (0.842 + 0.538i)23-s + (0.525 − 0.850i)25-s + (0.998 − 0.0598i)26-s + ⋯ |
L(s) = 1 | + (−0.525 − 0.850i)2-s + (−0.447 + 0.894i)4-s + (−0.873 + 0.486i)5-s + (0.995 − 0.0896i)8-s + (0.873 + 0.486i)10-s + (0.280 − 0.959i)11-s + (−0.473 + 0.880i)13-s + (−0.599 − 0.800i)16-s + (0.998 + 0.0598i)17-s + (−0.913 + 0.406i)19-s + (−0.0448 − 0.998i)20-s + (−0.963 + 0.266i)22-s + (0.842 + 0.538i)23-s + (0.525 − 0.850i)25-s + (0.998 − 0.0598i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03518260519 + 0.1462910001i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03518260519 + 0.1462910001i\) |
\(L(1)\) |
\(\approx\) |
\(0.5885785358 - 0.09359093363i\) |
\(L(1)\) |
\(\approx\) |
\(0.5885785358 - 0.09359093363i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.525 - 0.850i)T \) |
| 5 | \( 1 + (-0.873 + 0.486i)T \) |
| 11 | \( 1 + (0.280 - 0.959i)T \) |
| 13 | \( 1 + (-0.473 + 0.880i)T \) |
| 17 | \( 1 + (0.998 + 0.0598i)T \) |
| 19 | \( 1 + (-0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.842 + 0.538i)T \) |
| 29 | \( 1 + (-0.936 + 0.351i)T \) |
| 31 | \( 1 + (-0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.163 + 0.986i)T \) |
| 43 | \( 1 + (0.753 + 0.657i)T \) |
| 47 | \( 1 + (0.525 + 0.850i)T \) |
| 53 | \( 1 + (0.447 - 0.894i)T \) |
| 59 | \( 1 + (0.193 - 0.981i)T \) |
| 61 | \( 1 + (-0.887 - 0.460i)T \) |
| 67 | \( 1 + (-0.978 + 0.207i)T \) |
| 71 | \( 1 + (-0.936 - 0.351i)T \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.999 - 0.0299i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−17.0801276741922453053016678724, −16.97154501556269018841801584901, −16.199017802625485539990990760, −15.319725131669283734975573643756, −15.02293246175076715776820920541, −14.58997838963924832883798282283, −13.51673176446071452098451832443, −12.73839292641561110847308388314, −12.3673189019888866675510054509, −11.400366871672109940311755560537, −10.6266202225652032341786374335, −10.07096824624977813393409204592, −9.14995944637661174604989191520, −8.80597402110731067328195223833, −7.89702835328720206617248044688, −7.32450514765944362533852479003, −7.05076623853992267231641852100, −5.747937839989339538486298966049, −5.426366652917570534037153630493, −4.372380548452721013291801439756, −4.116754983816832977635805523791, −2.865375671737683030890472119759, −1.8566109426836761979512126757, −0.88847214958091350791993479860, −0.06092694622816439647637164245,
1.1208220909634998981313008349, 1.85030604098091748414790653862, 2.96291454331601045170990325767, 3.35247712947976747913389935825, 4.07841030906506423579992542196, 4.77903166380365228376748806533, 5.81255762486568178988942285767, 6.79545541795704597176878665717, 7.39058629106990256913220372417, 8.05221115200111269337273436783, 8.73873160378001364564201828619, 9.32030669221691961261172675146, 10.13539427952466210731466499416, 10.90577314230901327146232927209, 11.25017054024239323343492474479, 11.9458758986975906985318012049, 12.49493994972682970446016520795, 13.205493240556994307892619391408, 14.18088829070788877367721072493, 14.50796532349511873169563282129, 15.401779256563793034051762245433, 16.41170517806022794250378850659, 16.60885209685096565362837076356, 17.32575848589938131977277588087, 18.26617445422537479503894066840