L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999i·10-s + (−0.866 − 0.499i)12-s − 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (−0.499 + 0.866i)20-s + (0.499 − 0.866i)21-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999i·10-s + (−0.866 − 0.499i)12-s − 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (−0.499 + 0.866i)20-s + (0.499 − 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.276212739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276212739\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−10.22847689086900245240067040943, −9.570281839504335688484285424420, −8.606667575155535234664615223402, −7.17723765765442203687340013052, −6.78021881009993992671403625923, −5.81028633936786909154771350224, −5.52453209054353937105380945534, −4.23298361836153261841330256782, −3.37196237862551568776513028710, −2.37945671012028539417579573334,
0.948050536544041365989089782437, 2.10149099611464664625096464124, 3.51091774521443874665286719630, 4.54937628855373487676906133491, 5.38773238138527138026279814434, 5.98906037052576786508416166231, 6.85234005909318328479620538192, 7.58706063454325525774045507735, 9.101857948688342680507353097287, 9.716647113164145661304169904686