L(s) = 1 | + (0.809 + 0.587i)2-s + i·3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s − 9-s + (−0.309 + 0.951i)10-s + (−0.951 − 0.309i)11-s + (−0.951 + 0.309i)12-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + (−0.809 − 0.587i)18-s + (0.587 + 0.809i)19-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + i·3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.587 + 0.809i)6-s + (−0.309 + 0.951i)8-s − 9-s + (−0.309 + 0.951i)10-s + (−0.951 − 0.309i)11-s + (−0.951 + 0.309i)12-s + (0.587 − 0.809i)13-s + (−0.951 + 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + (−0.809 − 0.587i)18-s + (0.587 + 0.809i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.9350618331 + 1.717372637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.9350618331 + 1.717372637i\) |
\(L(1)\) |
\(\approx\) |
\(0.7717798044 + 1.200422678i\) |
\(L(1)\) |
\(\approx\) |
\(0.7717798044 + 1.200422678i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-0.951 - 0.309i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.587 + 0.809i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (0.951 - 0.309i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.951 + 0.309i)T \) |
| 71 | \( 1 + (-0.951 - 0.309i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.587 - 0.809i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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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
−24.24282646562050845676371245289, −23.94266774160452594695044221033, −23.11682032856320935670580593740, −21.926572866065170798457346364042, −21.0586684758378470314138118343, −20.133039763243079511881204392477, −19.5786340957894615853680207587, −18.363206972047111865077016845046, −17.68768404098010678752401102497, −16.29948593285286741468377962532, −15.405521897674435621285296656797, −13.95652395298514453285297655143, −13.43289313746787716070257800870, −12.72058231699594210286494394444, −11.83268677043688069888600250635, −10.94645289913950038628695143003, −9.55816660647735516833035152538, −8.55675995839385244205189864035, −7.22169341565455829282680750328, −6.097346143479331241958172419539, −5.23288104286312582568895679260, −4.10372210809343905007874482680, −2.49418055108941569634717033691, −1.68058640817009150586869404097, −0.41227602385267540015099758090,
2.61180667968559229032392791677, 3.29191054181739548610322182389, 4.487627428733055279703774912717, 5.611308271944046681593345252278, 6.3134868090658279561298637723, 7.70759447343141624661360426638, 8.63023582393047252886959094434, 10.14076357007636717560047476186, 10.82445728543810623467521032575, 11.835546541944721791052277535868, 13.25658834529220630305284138270, 14.041189620002678805257692471743, 14.87684749422680603109030963312, 15.76984626938922277574164000391, 16.24175688650929855476520229979, 17.655234935000399179933762908727, 18.22188247350123633209660228982, 19.912174702777444595835476611449, 20.87103589446352260990157442062, 21.49462823105954024839597332969, 22.55761708743366125047511131150, 22.77334322361940895204711411652, 23.96339632542312413649770929139, 25.19175823922031937967664178431, 25.85234688799019884749440943257