| 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
−10.49718899091404525502782160858, −10.14939663281111997653982561888, −8.907470676605050681520839461233, −8.640122336424060799242285811092, −7.33036099033575381441826313102, −6.43517030707162442565253190775, −5.25085908600039977139871308762, −4.21964194788725358658580735388, −2.93130979497482202946599239917, −1.92109155236737950928193837458,
0.44155870877094422270764707391, 2.05666363269315623948580046852, 3.27295994229956697237430734442, 5.00388013004078526659385553429, 5.90058905968155258316911307215, 6.63614127733524910894761682870, 7.930505567834895156601446499915, 8.043851809920267107388856481851, 9.412412342766300336033089431648, 9.902096609112803657617312240984
