L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.866 + 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.866 + 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.648098224 - 1.171499593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648098224 - 1.171499593i\) |
\(L(1)\) |
\(\approx\) |
\(1.252652047 - 0.5620247600i\) |
\(L(1)\) |
\(\approx\) |
\(1.252652047 - 0.5620247600i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 + (0.984 + 0.173i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.642 - 0.766i)T \) |
| 47 | \( 1 + (-0.342 + 0.939i)T \) |
| 53 | \( 1 + (0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.342 + 0.939i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.984 - 0.173i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−21.59040690881448824450331043386, −20.90609689902464301683287402063, −20.024967809187294137685895242843, −19.15045002980772540800019412318, −18.458038799398691306333739708318, −17.98797991384976209885086681167, −16.992834535970411182452100063275, −16.08778918130438548468436202764, −15.216068648777904089969235382900, −14.81248552598750923582584603389, −14.03988959433861486374526332644, −13.26326953992861162306908599433, −12.49595245860066498292219598031, −11.112761506786656957355788928329, −10.34564489454455125169427496334, −9.18510744551926550236846543472, −8.618694933021664594014080615081, −8.15440051451617679215866717727, −7.15920439111691204841998001576, −6.37458471587892230645131426634, −5.3366601445284215012672347979, −4.379871688316411200046853441994, −3.48465845707512708456566543618, −2.02874036867072474446245398504, −1.26880655695204122707736330798,
1.0246757400639495723463933546, 1.89979959182005734176720989251, 2.84980295301480082594722169491, 3.813977523793132925207980922493, 4.411507628086861124124476681641, 5.49443607372599580230112623605, 7.27570051686813060864404251401, 7.73187736055460697021870478105, 8.721702291356943559420112487366, 9.25070286145846619623230024751, 10.16668549648462873696643644813, 11.067372193888107811721189647897, 11.52121309819443569735293429047, 12.88297616773564795815673181178, 13.43231605457967599162600274060, 14.08451641074421581258319090921, 14.92248171557767143665954618361, 15.91453000350777983735953188816, 16.829741790881936920689960508506, 17.80746633065809828372393308243, 18.40652205268187673128567285626, 19.1609168150771971010043546788, 20.02085255919542386279862768670, 20.651829956011993781387592851039, 20.98056284147491737301316518769