L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)6-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.342 + 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − i·18-s + (−0.766 + 0.642i)21-s + (0.342 − 0.939i)22-s + ⋯ |
L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)6-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.342 + 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − i·18-s + (−0.766 + 0.642i)21-s + (0.342 − 0.939i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4613755541 + 1.118863780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4613755541 + 1.118863780i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628471379 + 0.8947027792i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628471379 + 0.8947027792i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.342 + 0.939i)T \) |
| 17 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 + (-0.984 - 0.173i)T \) |
| 59 | \( 1 + (0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.342 - 0.939i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.642 + 0.766i)T \) |
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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
−30.17489255578163552102347790865, −28.8395040291015925008050149180, −28.19743456067697092354124708846, −26.980736443128336286914839526645, −25.263370127442951427274162590259, −24.28197394004864847068766262523, −23.32311405771839304759916557859, −22.73753093181668350775962048681, −21.26193708313579333392934886895, −20.2757302988966591387888999459, −19.35799644647074602511840055316, −18.03983208723926646634929940953, −17.45384584050725774812556594662, −15.44168395529705695909069605216, −14.288656040568581873350453847619, −13.18632881375465876906202485129, −12.43552146535656688207250323616, −11.15510552389728883280769067516, −10.390753905127251014508943395731, −8.42143287650339736318873209063, −7.05878540124952157451459000101, −5.63197242176260525266943673540, −4.47467338510360657746672707477, −2.582050305691738308236706259, −1.243817846017556829317516916872,
2.93874583384424857520688300602, 4.462614262084576253186155237602, 5.30495526104295395605735922909, 6.54121470786308154771779866026, 8.27029694652662559418842753358, 9.181789455723081964598637600097, 11.09373443259902656918116993472, 11.79783501124933219577295176462, 13.50333022234825850104095018631, 14.53766436452267780312579526247, 15.54758466138180666598564630974, 16.361026485193741423788253668646, 17.42325383755805830550012112927, 18.5532318187199751904444772807, 20.61852429677635156781203762076, 21.3670653585992827678544820089, 22.08327729812060024947487020651, 23.32266574065283246233009588011, 24.15594121187379585589924828421, 25.305963390259519229109280676132, 26.57768711332397809360824173643, 27.10138776123849025065476342890, 28.415308252468844789013768933131, 29.58063091696048450932667079592, 31.242504856861151267300467541458