L(s) = 1 | + (0.235 − 0.971i)3-s + (−0.415 + 0.909i)4-s + (−0.928 − 0.371i)7-s + (−0.888 − 0.458i)9-s + (0.786 + 0.618i)12-s + (−1.32 − 1.04i)13-s + (−0.654 − 0.755i)16-s + (0.746 − 1.16i)19-s + (−0.580 + 0.814i)21-s + (−0.786 − 0.618i)25-s + (−0.654 + 0.755i)27-s + (0.723 − 0.690i)28-s + (−0.186 − 1.29i)31-s + (0.786 − 0.618i)36-s + 1.85·37-s + ⋯ |
L(s) = 1 | + (0.235 − 0.971i)3-s + (−0.415 + 0.909i)4-s + (−0.928 − 0.371i)7-s + (−0.888 − 0.458i)9-s + (0.786 + 0.618i)12-s + (−1.32 − 1.04i)13-s + (−0.654 − 0.755i)16-s + (0.746 − 1.16i)19-s + (−0.580 + 0.814i)21-s + (−0.786 − 0.618i)25-s + (−0.654 + 0.755i)27-s + (0.723 − 0.690i)28-s + (−0.186 − 1.29i)31-s + (0.786 − 0.618i)36-s + 1.85·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5782201490\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5782201490\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (0.928 + 0.371i)T \) |
| 67 | \( 1 + (0.928 - 0.371i)T \) |
good | 2 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 11 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (1.32 + 1.04i)T + (0.235 + 0.971i)T^{2} \) |
| 17 | \( 1 + (-0.327 - 0.945i)T^{2} \) |
| 19 | \( 1 + (-0.746 + 1.16i)T + (-0.415 - 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 - 1.85T + T^{2} \) |
| 41 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 43 | \( 1 + (1.80 - 0.822i)T + (0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.888 - 0.458i)T^{2} \) |
| 53 | \( 1 + (-0.327 + 0.945i)T^{2} \) |
| 59 | \( 1 + (0.723 + 0.690i)T^{2} \) |
| 61 | \( 1 + (-0.947 + 1.09i)T + (-0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 73 | \( 1 + (0.357 - 1.85i)T + (-0.928 - 0.371i)T^{2} \) |
| 79 | \( 1 + (0.459 - 0.584i)T + (-0.235 - 0.971i)T^{2} \) |
| 83 | \( 1 + (0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (-0.888 + 0.458i)T^{2} \) |
| 97 | \( 1 + (-0.235 + 0.408i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.581410739724046786982603981988, −8.441894505629287252191373827512, −7.72268596420200219255591764321, −7.25760834106067355669396579817, −6.40054653459018322017641236468, −5.33781076368097200159509849882, −4.19937472803547417993388808974, −3.04832100792922303820175144633, −2.55917015063685397446155118957, −0.44008669005818252623911333446,
1.96311762615408353584396952253, 3.19743107239729364482923068249, 4.17004759072200348752711109511, 5.06028313583295112784666751871, 5.72694736173881162336802181072, 6.61484348165345408715439146369, 7.70514677940321892407559905704, 8.885615985024382457858975395658, 9.366746412719987506261536360998, 9.982317149934772924092566579894