L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.623 + 0.781i)17-s + 18-s − 19-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + (0.222 − 0.974i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.623 + 0.781i)17-s + 18-s − 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3430338783 + 0.7425322474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3430338783 + 0.7425322474i\) |
\(L(1)\) |
\(\approx\) |
\(0.6381898100 + 0.3329432151i\) |
\(L(1)\) |
\(\approx\) |
\(0.6381898100 + 0.3329432151i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.623 + 0.781i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.623 + 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.900 - 0.433i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 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
−33.308707025842836179157827816272, −32.08792581785433304386675529324, −30.786828813113829115214818548790, −29.248852173955944215190399537594, −28.65469980739025150032462810622, −27.439005243846592102023208979520, −25.80782201065911640541746108261, −25.18996029158326816353397464534, −24.06053618772629862671926060270, −23.24359749257749210998054267726, −20.743979448449691130826902338024, −20.0438257629569380330635635655, −18.60833122611105934859990377817, −17.78711495775712956310234342298, −16.59544110489997644504699280495, −15.27475485215692547402810999317, −13.609956633092308939403326271100, −12.46071179133927016620165387661, −10.77939776993237889842398255019, −9.01461692871406346610692369953, −8.177389398440013871384527971223, −6.77759266732586358806650115353, −5.3273235709835300239371459186, −2.20600916638274366885652660232, −0.60056917268828823195767373265,
2.450817573925278222369971434790, 3.80439418237328399958543743528, 6.20955800913001390301304853844, 8.028222522368935241349244935339, 9.29711190686766062363123164485, 10.654754976158458194893196228553, 11.13461553563507124338930876297, 13.33379424749684009460849082121, 15.02067574958208697591472212300, 16.042946689603258902231496815826, 17.38213803410187116352609075318, 18.63016350166843136059947531571, 19.70407177138569428942382907089, 21.23512866805905211479305598942, 21.63830618840411725815625821403, 23.26883772544913077802971689232, 25.37862981546304768575928168896, 26.19104724397955325874576107632, 26.8859720372463498207033002235, 28.12129119324467179153034054961, 29.12325160225205358370548240480, 30.45814844011693481021889476712, 31.4456605407810626394234552898, 33.17012068403343072753103560118, 33.98836440294976044649814766948