| L(s) = 1 | − i·7-s − 11-s − i·13-s − i·17-s − i·23-s + 29-s + 31-s + i·37-s + 41-s − i·43-s + i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯ |
| L(s) = 1 | − i·7-s − 11-s − i·13-s − i·17-s − i·23-s + 29-s + 31-s + i·37-s + 41-s − i·43-s + i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2671025905 - 0.9402403883i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2671025905 - 0.9402403883i\) |
| \(L(1)\) |
\(\approx\) |
\(0.8741122126 - 0.3015792911i\) |
| \(L(1)\) |
\(\approx\) |
\(0.8741122126 - 0.3015792911i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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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
−19.70509193487065953600537724628, −19.265178323666083364080180966238, −18.53706337236614997238295272526, −17.84726354521193835755544721802, −17.20589445984302019377770538576, −16.106641631569318201825381460835, −15.77188534605314930451134656051, −14.94330466431262489477741540155, −14.25443296219658052414350936262, −13.35421351382697446798197519638, −12.70561446125248346257842746968, −11.934646373917492757704920324801, −11.3042391920237257997479874917, −10.4069824137243992767563435877, −9.65805115781563563508291003676, −8.842202286544896163352646402225, −8.20934706878384260195465965726, −7.39716761092503866270713572058, −6.368515565687763926005713721410, −5.76636043991341458189997072854, −4.921967997855210394073486413551, −4.09319144592521624977085792718, −3.00287789375905457271385868296, −2.27866004138189643436409992655, −1.392942345019864396125221856793,
0.33630030113080020280960583733, 1.22503895261987560102383253698, 2.69957165978987365947907131132, 3.056182050185884466897084144802, 4.40184870837897184289668267315, 4.838886896396302414837803655770, 5.87562212165941190698598671735, 6.71303709530108057976792816427, 7.62202212060566163174373799900, 8.02760949650893290676098236222, 9.02898804876473882168475309311, 10.10631794599702135751313693183, 10.431460272470891456307003258576, 11.173432358024209452685435585149, 12.219846677100394650834782339661, 12.82957555385839569744799446910, 13.74155055742398670528714757721, 14.02555037752148638990337667126, 15.20795335337350666901174317364, 15.731812204723078925618740285, 16.49488754752675950284257044305, 17.239635817510196409101762073554, 17.951316969209848990080775227387, 18.53431382579728111254890119097, 19.4315264378347369084787140458
