| L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (0.415 + 0.909i)6-s + (−0.525 − 0.605i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−3.81 − 2.45i)11-s + (−0.841 − 0.540i)12-s + (1.33 − 1.54i)13-s + (0.769 + 0.225i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−2.27 − 4.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (0.429 − 0.125i)5-s + (0.169 + 0.371i)6-s + (−0.198 − 0.229i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.207 + 0.238i)10-s + (−1.14 − 0.738i)11-s + (−0.242 − 0.156i)12-s + (0.370 − 0.427i)13-s + (0.205 + 0.0603i)14-s + (−0.0367 − 0.255i)15-s + (−0.163 − 0.188i)16-s + (−0.551 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.359618 - 0.648067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.359618 - 0.648067i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-0.259 - 4.78i)T \) |
| good | 7 | \( 1 + (0.525 + 0.605i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (3.81 + 2.45i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 1.54i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.27 + 4.97i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (1.39 - 3.06i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.36 + 5.17i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.15 + 8.05i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-4.13 - 1.21i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (5.00 - 1.47i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.78 - 12.4i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + (6.06 + 7.00i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.99 + 2.30i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (2.18 + 15.1i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 7.38i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (4.84 - 3.11i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.55 + 3.40i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.29 + 1.49i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.51 - 0.737i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.999 - 6.94i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.760 + 0.223i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.902096609112803657617312240984, −9.412412342766300336033089431648, −8.043851809920267107388856481851, −7.930505567834895156601446499915, −6.63614127733524910894761682870, −5.90058905968155258316911307215, −5.00388013004078526659385553429, −3.27295994229956697237430734442, −2.05666363269315623948580046852, −0.44155870877094422270764707391,
1.92109155236737950928193837458, 2.93130979497482202946599239917, 4.21964194788725358658580735388, 5.25085908600039977139871308762, 6.43517030707162442565253190775, 7.33036099033575381441826313102, 8.640122336424060799242285811092, 8.907470676605050681520839461233, 10.14939663281111997653982561888, 10.49718899091404525502782160858
