L(s) = 1 | + (0.5 − 0.866i)3-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 13-s + (1 − 1.73i)17-s + (−2.5 − 4.33i)19-s + (−2 + 1.73i)21-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 0.999·27-s − 8·29-s + (1.5 − 2.59i)31-s + (−0.999 − 1.73i)33-s + (4.5 + 7.79i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 0.277·13-s + (0.242 − 0.420i)17-s + (−0.573 − 0.993i)19-s + (−0.436 + 0.377i)21-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 0.192·27-s − 1.48·29-s + (0.269 − 0.466i)31-s + (−0.174 − 0.301i)33-s + (0.739 + 1.28i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.665899 - 1.00107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.665899 - 1.00107i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−10.18987665732156992179273476391, −9.289119926915806724691452792594, −8.547433564021256449521646652138, −7.57842880539696012350789495096, −6.58152171388148417110033479455, −6.08374990537256173628167383456, −4.57356004989035895150065611371, −3.45402249761594588939019684739, −2.41844532142829202541071407104, −0.60812738537787409802359400123,
1.89900277113488870219400588468, 3.35911810172884238495230737964, 4.03603809203216524183816860156, 5.44262333822350199102950112321, 6.20206266565479074778989239588, 7.32630130891329094292412796946, 8.245822510331323119501921809955, 9.355376221145531575895543091884, 9.660380412515403207056694063490, 10.67769103457568103418758291920