L(s) = 1 | + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 7-s + (0.900 + 0.433i)8-s + (−0.900 + 0.433i)10-s + (0.222 − 0.974i)11-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + (0.900 − 0.433i)17-s + (−0.222 − 0.974i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.222 − 0.974i)23-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 7-s + (0.900 + 0.433i)8-s + (−0.900 + 0.433i)10-s + (0.222 − 0.974i)11-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + (0.900 − 0.433i)17-s + (−0.222 − 0.974i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.222 − 0.974i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.616033921 + 0.1870955074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616033921 + 0.1870955074i\) |
\(L(1)\) |
\(\approx\) |
\(1.029832489 + 0.2145966697i\) |
\(L(1)\) |
\(\approx\) |
\(1.029832489 + 0.2145966697i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (-0.623 + 0.781i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + (0.900 - 0.433i)T \) |
| 19 | \( 1 + (-0.222 - 0.974i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 29 | \( 1 + (-0.623 + 0.781i)T \) |
| 31 | \( 1 + (0.623 - 0.781i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 + (0.900 - 0.433i)T \) |
| 61 | \( 1 + (0.623 + 0.781i)T \) |
| 67 | \( 1 + (-0.222 - 0.974i)T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (-0.900 - 0.433i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (-0.623 - 0.781i)T \) |
| 97 | \( 1 + (-0.222 + 0.974i)T \) |
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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
−28.39854057327208709133529551859, −27.678810659341374318640709001175, −26.69944216517144342561127157837, −25.473104157477184787240538351171, −24.83802575258026099346815865483, −23.39727405779111087116532576661, −22.00634819871238810818785135235, −21.16669020427632491003007245325, −20.5438120932773623419855278870, −19.36631624287544643507970547938, −18.18603106378656041501029424119, −17.30054646400707892218595196889, −16.762620493836608961048052619617, −14.874129055292618565380612028194, −13.79127062567440634138224507184, −12.504649661241041710279832481475, −11.752288273477654745462009542796, −10.26630693216135368381327453414, −9.594590849636385585945928626711, −8.3596882740364917769287378748, −7.24742866042267803367210569153, −5.35226187507210662573679993730, −4.141039243887400920788461980234, −2.21975975589602503755316415133, −1.371038252499007130902923608265,
0.933531392474831685974139767, 2.534390413877834785973825224163, 4.86429270448090103888958000599, 5.81604128977806452578214800932, 7.04391765779169154505738551967, 8.18559632195301585423902954932, 9.32219269961366614717112212283, 10.41040001271665545046015405489, 11.39220476642680141167836077106, 13.30499763595741418001918928707, 14.40319500688996005414888611085, 14.907621425406411482415023656853, 16.51206322486857169244390502848, 17.286515860710764091362668460092, 18.182328917685134821754543807390, 19.01949837905657241030559981930, 20.373917779643710969910611664111, 21.58968058100168220351613316779, 22.55337321512655653756783176866, 23.96341877384156328622995857005, 24.62545097783314760913362035514, 25.51119263910882154719438839386, 26.62158872945276778057106639742, 27.27300176221597957485359818086, 28.35766846226878881205441588002