L(s) = 1 | + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.850 + 0.526i)5-s + (0.0922 − 0.995i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + (−0.982 + 0.183i)10-s + (−0.850 − 0.526i)11-s + (0.445 − 0.895i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.739 + 0.673i)15-s + (0.0922 + 0.995i)16-s + (−0.273 − 0.961i)17-s + ⋯ |
L(s) = 1 | + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.850 + 0.526i)5-s + (0.0922 − 0.995i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + (−0.982 + 0.183i)10-s + (−0.850 − 0.526i)11-s + (0.445 − 0.895i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.739 + 0.673i)15-s + (0.0922 + 0.995i)16-s + (−0.273 − 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.217047995 - 1.087268614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.217047995 - 1.087268614i\) |
\(L(1)\) |
\(\approx\) |
\(1.332265167 - 0.3114676968i\) |
\(L(1)\) |
\(\approx\) |
\(1.332265167 - 0.3114676968i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 647 | \( 1 \) |
good | 2 | \( 1 + (0.932 + 0.361i)T \) |
| 3 | \( 1 + (-0.273 - 0.961i)T \) |
| 5 | \( 1 + (-0.850 + 0.526i)T \) |
| 7 | \( 1 + (0.445 - 0.895i)T \) |
| 11 | \( 1 + (-0.850 - 0.526i)T \) |
| 13 | \( 1 + (0.445 - 0.895i)T \) |
| 17 | \( 1 + (-0.273 - 0.961i)T \) |
| 19 | \( 1 + (0.0922 + 0.995i)T \) |
| 23 | \( 1 + (0.0922 - 0.995i)T \) |
| 29 | \( 1 + (0.0922 - 0.995i)T \) |
| 31 | \( 1 + (-0.850 - 0.526i)T \) |
| 37 | \( 1 + (0.0922 - 0.995i)T \) |
| 41 | \( 1 + (0.445 + 0.895i)T \) |
| 43 | \( 1 + (0.932 - 0.361i)T \) |
| 47 | \( 1 + (-0.273 - 0.961i)T \) |
| 53 | \( 1 + (0.0922 - 0.995i)T \) |
| 59 | \( 1 + (-0.982 + 0.183i)T \) |
| 61 | \( 1 + (-0.273 + 0.961i)T \) |
| 67 | \( 1 + (-0.273 + 0.961i)T \) |
| 71 | \( 1 + (0.0922 - 0.995i)T \) |
| 73 | \( 1 + (0.932 + 0.361i)T \) |
| 79 | \( 1 + (0.932 - 0.361i)T \) |
| 83 | \( 1 + (-0.982 - 0.183i)T \) |
| 89 | \( 1 + (0.932 + 0.361i)T \) |
| 97 | \( 1 + (-0.982 + 0.183i)T \) |
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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
−23.08125203384461684535571033056, −22.02340430006625036735526843871, −21.497873676602443191767207307, −20.844060295099003587049482329839, −20.05232199568323277616949869406, −19.292721160317595527163483171743, −18.20761118260210506328973013376, −17.053366807786124749727183426724, −15.94144020223002881748415106995, −15.571916626332245774174008817218, −14.9833328630288896041997958471, −13.98873527629207453168098355541, −12.73857620960101495382905165737, −12.18414123803550192761411139475, −11.12844990398435399693846375698, −10.927379462180120126854630094, −9.466393920516409464058681681833, −8.772281649993840578405838296926, −7.541338399481966726793835269336, −6.2591211487194899183713620607, −5.210308933978785452697672104530, −4.73289262573034577407857703360, −3.84639467928728183239921657186, −2.859888809922093546017303429888, −1.56627622989316250900883328892,
0.60351449111619664291522361927, 2.29325469112349625569193262172, 3.211183748733638770122635131756, 4.21769611523727213342433120720, 5.341524312657144555226803127473, 6.20745370150875122668750878482, 7.220333964225160560746687028170, 7.78362657696462690926142651892, 8.320138711453534878954957456275, 10.55582184320545494879163173851, 11.0465336857326497668112624722, 11.842236493864695019273716690066, 12.77337942656179582828328879819, 13.49018867134532210463999364237, 14.22734389467200872708576713127, 15.014856024337297701862246512116, 16.129798259321762874558260874009, 16.627662212263023786876812747746, 17.86712206506834270904558807857, 18.40986016368668315039833737899, 19.5138398382754842890362748804, 20.35578559191349053328338361461, 20.93380388231210481341904670036, 22.34379643950372090354029959872, 23.05914988923553683882699001774