L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (−0.433 + 0.900i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.781 − 0.623i)11-s + 12-s + (−0.781 − 0.623i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (−0.433 − 0.900i)19-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (−0.433 + 0.900i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.781 − 0.623i)11-s + 12-s + (−0.781 − 0.623i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (−0.433 − 0.900i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.003296360207 + 0.005654153551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003296360207 + 0.005654153551i\) |
\(L(1)\) |
\(\approx\) |
\(0.4698207215 + 0.2068819421i\) |
\(L(1)\) |
\(\approx\) |
\(0.4698207215 + 0.2068819421i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 3 | \( 1 + (-0.900 - 0.433i)T \) |
| 7 | \( 1 + (-0.433 + 0.900i)T \) |
| 11 | \( 1 + (-0.781 - 0.623i)T \) |
| 13 | \( 1 + (-0.781 - 0.623i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.433 - 0.900i)T \) |
| 23 | \( 1 + (-0.974 - 0.222i)T \) |
| 31 | \( 1 + (-0.974 + 0.222i)T \) |
| 37 | \( 1 + (0.623 + 0.781i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.974 - 0.222i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.433 + 0.900i)T \) |
| 67 | \( 1 + (-0.781 + 0.623i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.222 - 0.974i)T \) |
| 79 | \( 1 + (-0.781 + 0.623i)T \) |
| 83 | \( 1 + (-0.433 - 0.900i)T \) |
| 89 | \( 1 + (0.974 - 0.222i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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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
−27.73998536542167441843815925718, −26.8646326183065482647446102944, −26.084325174217832606106380567539, −24.06095094250140900688221424205, −23.39162648564561076986406383922, −22.51447148681430281862853921136, −21.716600548273535484469066505080, −20.662038968758855175000283478881, −19.8401710785456528755374331454, −18.568555093983623783035482886767, −17.5988363111998347192975377164, −16.67498880640022537041599919229, −15.401543277607198021024756278059, −14.11886233781630065139339422451, −12.89474355921678891875291552315, −12.1181087887197165070712177185, −10.873230067720517269982636687575, −10.20872820709487158067694153920, −9.283749731664214647565738099764, −7.361645646299927390513966269214, −5.92394172774822429548235855965, −4.61844825710024065978706110230, −3.85475783163396798686705159912, −2.01863630420393975823420670240, −0.005593578959840346925138511024,
2.67230759497871986334108559171, 4.646284429396466124149092331094, 5.62899495166970051590780103659, 6.46351635937988098982367678144, 7.64786630793781181171034107875, 8.7916966524430211674981220046, 10.20104411662812520800072224170, 11.64645499557519084027112832359, 12.79786068812961872508489788874, 13.36407170971656560172700618868, 14.99882124064239716070904621585, 15.84678189857671809869351866320, 16.703286398333307201384054444683, 17.89637358653492637122892300226, 18.39873199131000442480237065616, 19.64179252763063619213611897021, 21.751981829693256230563928586194, 21.98933853696561688837179992732, 23.12181660665849865616667199499, 24.10973626886231834669192090283, 24.68335725805336028521534789595, 25.77657277697414869639332998521, 26.852093974739122675796920700575, 27.916255824556999249559331942006, 28.77239512978149509850129510449