L(s) = 1 | + 0.517·5-s + (0.866 − 0.5i)13-s + (1.67 + 0.965i)17-s − 0.732·25-s + (−1.67 + 0.965i)29-s + (1.5 − 0.866i)37-s + (0.258 + 0.448i)41-s + (−0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.448 − 0.258i)65-s + i·73-s + (0.866 + 0.499i)85-s + (0.707 + 1.22i)89-s + (1.73 + i)97-s + ⋯ |
L(s) = 1 | + 0.517·5-s + (0.866 − 0.5i)13-s + (1.67 + 0.965i)17-s − 0.732·25-s + (−1.67 + 0.965i)29-s + (1.5 − 0.866i)37-s + (0.258 + 0.448i)41-s + (−0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.448 − 0.258i)65-s + i·73-s + (0.866 + 0.499i)85-s + (0.707 + 1.22i)89-s + (1.73 + i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.530009965\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530009965\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 - 0.517T + T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 0.517iT - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.631262262596583114589580069693, −7.958771787002105490510112953919, −7.37277431905176974379546025771, −6.24929668905232549805560323085, −5.78072401225649609387975020133, −5.15419006239725740822202104797, −3.84966105601654730415197075797, −3.40313876373107332395890197335, −2.13395882685487216193525138293, −1.19275030427299492272946072231,
1.12310692478247259142339992122, 2.17859077986770378597526292831, 3.21550868631154813511820473576, 4.02015969861247860335737717910, 4.97737372285398679915660006430, 5.91005595911086490354529832731, 6.16736959086043832302306796245, 7.46183279733293350979282022048, 7.73134109596296483809018325989, 8.785786243361850596281900316598