L(s) = 1 | + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯ |
L(s) = 1 | + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.112534816 + 0.3911299016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112534816 + 0.3911299016i\) |
\(L(1)\) |
\(\approx\) |
\(1.156413946 + 0.3261377696i\) |
\(L(1)\) |
\(\approx\) |
\(1.156413946 + 0.3261377696i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + iT \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−30.133557892162467253772323918999, −29.34404558439472436113021556355, −27.96565411404875538913149614266, −27.60325219323456630340671673864, −26.206347375941238496610722937768, −25.37695292107608939342650291379, −23.88603959359456724491022059489, −22.237400471490481454908192741, −21.7094479314278484467045440937, −20.73543214795237476523145956968, −19.931324488345658577386011190235, −18.79376463880898416007120630382, −17.31374070350886971238514324418, −16.47038241305807360692028205802, −14.63800665301335706512090638775, −13.88453856479112139835976916850, −12.70934661965771298415299880271, −11.27824495308560913818817399333, −10.09619817142799019299818655887, −9.27197911243815642125971772914, −8.34063961243424624274279936082, −5.77394266120280268608953370886, −4.524514783920292227980988675094, −3.26634892294220613975896594681, −1.75830341764648968613489359920,
1.79428331052860771971284115703, 3.70711876351360264046503423456, 5.731447481157229711600216361736, 6.6327175674274969556143801245, 7.68981771894598556102375197413, 8.990821895152488903760020919548, 10.04527747971238999985139031744, 12.18130455106352443620670475528, 13.30551152423649127486178517117, 14.3479642233397157986987253519, 14.8583105844581390430183092569, 16.689279422142645868732274508469, 17.63664048050657853467642795171, 18.47315620065466373876004735078, 19.48458224622562189993594717496, 21.076855159753270406538826325227, 22.3742027474005171526206848548, 23.32478803013400650361668926186, 24.46679003901237006979859781149, 25.42841075781744173820194565634, 25.77629930740802395288805724881, 27.07247691419995534857983773900, 28.381523947790246233939539962185, 29.864373965940021778743060274094, 30.42662450524056725339077179878