L(s) = 1 | + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.766 − 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + 18-s + (−0.173 + 0.984i)21-s + (0.766 + 0.642i)22-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.766 − 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s + 18-s + (−0.173 + 0.984i)21-s + (0.766 + 0.642i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.099023001 + 0.3694081111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.099023001 + 0.3694081111i\) |
\(L(1)\) |
\(\approx\) |
\(1.421858729 - 0.07750006576i\) |
\(L(1)\) |
\(\approx\) |
\(1.421858729 - 0.07750006576i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.173 + 0.984i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.766 + 0.642i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
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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
−30.19847810436809557943029428003, −29.04037156115503409330219332046, −27.259845066593016378359594451465, −26.53827775388278389424962944709, −25.67918837245098955647092305039, −24.544878077546222347458524030248, −23.84736796551711123793995926542, −23.01168598929626473220810600586, −21.37142635254126394704349836757, −20.427325668844907664164168228932, −18.89341701603248496858016829551, −18.18703861297390185530206286029, −16.90771705222741417844927226795, −15.81956902195692515379194373387, −14.474380005648325407019843357210, −13.75787750010725685825650251115, −12.96381541936076682901105833801, −11.21973086763165950301861306334, −9.36622231922939844268857520005, −8.29909850916286147682538216432, −7.37012175533149548442046968713, −6.28265109558496556166590415692, −4.5764281560229992446386055533, −3.18027835745301044818227633007, −0.911379565306398928369655305141,
1.83359593798650760732477226205, 3.02004940579589628025304414641, 4.3925302826580647141317653687, 5.53261961734548365616270152572, 8.05265677472669380825734932288, 8.94932532310373905004733394107, 10.13368708091746674072581395485, 11.099150073308508716257034213498, 12.53546900750803503576565365498, 13.53389267626831317783201045788, 14.8953541901138806899684492633, 15.46685897399951448647871510857, 17.44651211523117486839891275732, 18.55208424007533454130299909846, 19.579615082551631265369458894182, 20.69463610756062607529241886508, 21.25293886883521270073841680439, 22.291743288070841142520196965430, 23.4339731370685970355333315390, 24.95158505286936344116618621045, 25.97200827657479229387217679715, 27.14852620911129502305723306102, 28.01565728182748669020159934548, 28.70354483705808549040918192103, 30.541182797504876964040066966998