L(s) = 1 | + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.294i)2-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.988 − 0.149i)10-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.0747 + 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.051062345 - 0.8199747729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051062345 - 0.8199747729i\) |
\(L(1)\) |
\(\approx\) |
\(0.8526311799 - 0.2639495770i\) |
\(L(1)\) |
\(\approx\) |
\(0.8526311799 - 0.2639495770i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.955 - 0.294i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 11 | \( 1 + (0.733 - 0.680i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.0747 + 0.997i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.826 - 0.563i)T \) |
| 41 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.955 - 0.294i)T \) |
| 53 | \( 1 + (-0.826 - 0.563i)T \) |
| 59 | \( 1 + (0.988 + 0.149i)T \) |
| 61 | \( 1 + (0.826 - 0.563i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (0.955 - 0.294i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.733 + 0.680i)T \) |
| 97 | \( 1 + T \) |
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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
−28.34118358858319968553453868772, −27.0252064024115415457859798101, −26.10800402544657070037656384977, −25.409547180830587068211475135082, −24.506682808692842736645666052973, −23.50429371993674295430226320625, −22.02346542782454471939491835718, −21.15348351424289233839559905814, −19.955839725557711867162799957951, −19.08439549848989455306546289991, −17.91802142963800510343625150447, −17.245445038713599739058216440429, −16.38295167918064952702201822390, −14.93812324071449385463767987913, −14.2552871847527918403192592664, −12.73365716017462529620358554194, −11.39674418871545732425992778447, −10.25529103202656493652372677840, −9.406707305394609470295259290265, −8.45581523974870154086764870567, −6.82593328574102083387538766575, −6.29441646099190003968884422359, −4.65690106539982931004680089352, −2.480544952957142468116320412760, −1.39976812497220240151777583506,
0.74054913408618869703803120131, 2.11014056436689785117807937563, 3.45336216316290617515505980408, 5.50233763333004120827075603076, 6.60955496504032508564285942150, 7.96275431850179195531294534803, 9.107891161470619655337779792635, 9.9276353620852368510972774981, 10.9827482953430284952416097073, 12.16674157200438583423129894828, 13.30021958849838380432594965839, 14.53473368681223609304555074922, 15.95190412823166289051095911988, 16.92232304963314557686537880008, 17.70304501247134924442704120534, 18.616478973150124421062058408378, 19.74368310409177107346526909438, 20.66769428780160403735694338174, 21.57932277851227184501214925051, 22.4944199671103818633347575972, 24.201963489547489485076075836404, 25.18340820725337702579069484315, 25.57841870505013379600724487962, 27.06568227934701003720520290906, 27.51583675388052480219786994519