L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.415 − 0.909i)20-s + 22-s + (−0.142 + 0.989i)25-s + (−0.415 − 0.909i)26-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.415 − 0.909i)20-s + 22-s + (−0.142 + 0.989i)25-s + (−0.415 − 0.909i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.866454197 - 0.3945266845i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.866454197 - 0.3945266845i\) |
\(L(1)\) |
\(\approx\) |
\(1.278054093 - 0.3325360116i\) |
\(L(1)\) |
\(\approx\) |
\(1.278054093 - 0.3325360116i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 11 | \( 1 + (0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.841 - 0.540i)T \) |
| 17 | \( 1 + (0.959 - 0.281i)T \) |
| 19 | \( 1 + (-0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + (-0.654 + 0.755i)T \) |
| 41 | \( 1 + (0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (-0.841 + 0.540i)T \) |
| 61 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
| 71 | \( 1 + (0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.959 - 0.281i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.654 - 0.755i)T \) |
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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
−32.02551036258767987955263500398, −30.756568813317556501644448247211, −29.59617549680169795508216900701, −28.09425266749580764288728460253, −27.20415904138823323728844767038, −25.99247708309127300942019898689, −24.990812837243420767164904472581, −24.0056100853114793787460306595, −23.28177675000323371157125708855, −21.52234750624693352380605482046, −20.995375516782376788535757680494, −19.110960861977109756433741233535, −17.77583349043345991378828444834, −16.8813076745988220387243045382, −16.02845143613447761584026797208, −14.30988162722948778276053294135, −13.74152604938944844057319276912, −12.37618846557227332907971559210, −10.55466371234378343679850656155, −8.91403354693090072723697954303, −8.12241314210404600925653392981, −6.416601459670494923902922097971, −5.27158071707526897012188628328, −3.94742514485253257020278077870, −1.14637279172890918341549447822,
1.616964021996128329592665960060, 2.90277147030864019960691660893, 4.679088108887632028373671651816, 6.079874528213843794984437209933, 8.10392010855262101208130412879, 9.58133421393508575181185798995, 10.63283692803735190672451163535, 11.73645701036607403766832484815, 13.02480538348122589652654030834, 14.29164178945214000122888552096, 15.13636242238605123500603816486, 17.44585933062640840451712412938, 18.12238774917979865944017843596, 19.14871271678705057376008537758, 20.69744829897548164986541152244, 21.3217078374173922200699247685, 22.51744318824045906605779355363, 23.3894231080556825222442222107, 25.07560778283130896942258106282, 26.117403065345496775423845793809, 27.593848063826330870314724063629, 28.1628772417505416653411303463, 29.58146037191411701417713552197, 30.36769956193914399653244321212, 31.11502514216777583670866260731