L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + 12-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.507259678 - 0.7250414336i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.507259678 - 0.7250414336i\) |
\(L(1)\) |
\(\approx\) |
\(1.321806578 + 0.2895089615i\) |
\(L(1)\) |
\(\approx\) |
\(1.321806578 + 0.2895089615i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−20.61865231623078678445018450531, −20.035732732787522022208993908013, −19.579700071643061722902075065130, −18.849366575522735004071253080066, −17.35575336605329291992247245642, −16.67230866081791778359488918147, −15.99399241773836454753326515595, −15.17909854487593833654851921274, −14.55791176304423546719509609887, −13.902203667987562262300295013639, −13.050608301857440200729811748166, −12.12535492730805200931648964189, −11.44640696343874811348080213208, −10.744856240391585647716384657961, −9.978259495615067144041906350553, −9.275471211830001615265540182852, −8.20426657008109123010322381649, −7.4151405024242970029341777232, −6.279652634332560010852200116242, −5.00733467176307492668410809121, −4.64277491556005414050648033268, −3.90438576830760764670782740572, −3.23891248135889209183313870856, −2.06406785847048954169737778369, −0.88690152700682131278473453814,
0.25251277498596215309906814132, 1.983629869491670574046998731268, 2.73359517243152802288064933552, 3.43485595165278465468001186513, 4.50491324075982547573611834044, 5.664593994813662165360339352336, 6.126380917666594293103152139799, 7.222365292405260639515906668999, 7.82072533772325805207705879063, 8.21032622761558909211910402940, 9.319741208860984046603191990644, 10.66834022325525746466433603329, 11.779135920778086281235285206451, 12.16635040190804699325549663565, 12.58110943545791059753235703890, 13.83037596049657721379334077295, 14.40361532446017458340544777522, 14.936889728247357373939465868337, 15.6669278602384571339335532185, 16.46728059414604769555530119674, 17.63717956352971460031960367724, 18.0220770168389593289402486596, 19.05347865303558438568950231114, 19.46447963511204130425844990739, 20.60869187221575723761874433361