L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1 + 1.73i)23-s + 0.999·27-s + (0.5 − 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (0.499 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.499 + 0.866i)20-s + (−1 + 1.73i)23-s + 0.999·27-s + (0.5 − 0.866i)31-s + 0.999·33-s + 0.999·36-s − 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4774195106\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4774195106\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−14.04537565277773264555820701746, −13.21712899441185516945422273870, −12.17300394541636825118828706018, −11.18660620224420896302432564705, −9.771058824857733952186506670259, −8.995618197728112279657030086600, −7.900036263668485527728149050920, −5.87745731392387153437427664513, −4.83958004195883381465049249391, −3.48574567035271635281071490245,
2.19749917619495695658025058344, 4.86750891631774348547014659670, 6.13008093900409168117416047022, 6.97978145828428605996788480269, 8.495025006313518922315157726615, 10.16722206896139673314314510775, 10.56827862765709099156355649704, 12.03143578339260485688789095758, 13.09202432732099513747878414976, 14.08413891845906402596993050808