L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 7-s + 8-s + (−0.5 + 0.866i)9-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)21-s + (−0.5 − 0.866i)22-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 7-s + 8-s + (−0.5 + 0.866i)9-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 18-s + (0.5 + 0.866i)21-s + (−0.5 − 0.866i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7493325977 - 0.1619510970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7493325977 - 0.1619510970i\) |
\(L(1)\) |
\(\approx\) |
\(0.5871051878 - 0.2662138030i\) |
\(L(1)\) |
\(\approx\) |
\(0.5871051878 - 0.2662138030i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−29.58405930363759882008322444684, −28.778452762143196522714278020939, −27.42091028249805495963153496881, −27.22069767797567502113412896233, −25.758261657386634641099952199355, −25.13997025493791388024563756041, −23.63422119695295554647229646289, −22.640610059556049767791609854250, −22.0515037880881388322529285260, −20.27872269258700822284501091280, −19.31351249207235006511626653061, −17.90256257262080779985128565353, −16.92688130404687464087442286197, −16.13494534404999995668951490903, −15.196946149288540004136961562881, −14.11580275800595363554161141214, −12.47033963688025007383484740907, −10.91055972367194092521789613937, −9.711674132828103668734264782389, −9.118433287416195014042408156910, −7.30923577058409387829414276624, −6.11573377234112734838858333664, −5.05189825357740803548480255399, −3.48658840108925729744621219309, −0.55805807764059027320931887862,
1.060958596821136263492070983440, 2.550530449512920603019314588028, 4.18930032711182666460589212286, 6.20171853055659209316800961221, 7.32504309729048522299432997256, 8.79022316154758060202808409492, 9.957405876840713118717771544893, 11.29525380683775317781212120902, 12.27556202285972215033199254752, 13.03576887056176117008546252422, 14.32218177708608255235017633188, 16.55723773771032536640535227131, 17.026946294358975562711379925390, 18.3887552136322795391384216772, 19.286058874785705998691599616115, 19.83373899034294888240598593538, 21.53831678171901603493006431512, 22.41049614845421594339612838109, 23.33496306201211115476656486483, 24.78590510717837387987223039755, 25.76949348832090165346543984798, 26.89997357630257977515182690369, 28.20626795668210430280522992642, 28.83894445812640638535875266831, 29.74966261505731444281302099449