| L(s) = 1 | + (0.766 − 0.642i)2-s + (−1.07 − 1.35i)3-s + (0.173 − 0.984i)4-s + (0.620 − 3.51i)5-s + (−1.69 − 0.344i)6-s + (−0.670 − 2.55i)7-s + (−0.500 − 0.866i)8-s + (−0.673 + 2.92i)9-s + (−1.78 − 3.09i)10-s + (0.735 + 4.17i)11-s + (−1.52 + 0.826i)12-s + (−0.585 + 3.32i)13-s + (−2.15 − 1.52i)14-s + (−5.43 + 2.95i)15-s + (−0.939 − 0.342i)16-s + (1.28 + 2.22i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.622 − 0.782i)3-s + (0.0868 − 0.492i)4-s + (0.277 − 1.57i)5-s + (−0.692 − 0.140i)6-s + (−0.253 − 0.967i)7-s + (−0.176 − 0.306i)8-s + (−0.224 + 0.974i)9-s + (−0.565 − 0.978i)10-s + (0.221 + 1.25i)11-s + (−0.439 + 0.238i)12-s + (−0.162 + 0.921i)13-s + (−0.576 − 0.408i)14-s + (−1.40 + 0.762i)15-s + (−0.234 − 0.0855i)16-s + (0.311 + 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.242405 - 1.38032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.242405 - 1.38032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (1.07 + 1.35i)T \) |
| 7 | \( 1 + (0.670 + 2.55i)T \) |
| good | 5 | \( 1 + (-0.620 + 3.51i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.735 - 4.17i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.585 - 3.32i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.28 - 2.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.01 + 5.22i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.75 + 1.47i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.315 + 1.79i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 10.1i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + (1.32 - 7.49i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.13 - 1.79i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.550 + 3.12i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.04 + 7.00i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.98 - 2.17i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.904 - 5.13i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.43 + 1.20i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.31 - 5.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + (-2.65 + 2.22i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.275 - 1.56i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.58 - 4.47i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.6 + 12.3i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.37970843235972056138818447623, −9.990296460186685103458730639342, −9.416224891193765207068341199866, −8.015114478260534030834561662480, −7.02247502608740978329582862807, −6.03106951577869099777441080740, −4.78819424965537875778126361242, −4.32057251241345459139757532307, −2.05671408845477243514354254897, −0.894100143305431941610956927342,
3.00112384742574856071489338162, 3.47007420372313535142892967989, 5.33129495148294699943813205890, 5.89082287600839268803993902842, 6.64189382442354672744218691630, 7.896686523032352722826304398110, 9.130758936774411555429822222692, 10.16225740158305810826993507732, 10.85117452394487946708323878989, 11.74680778808824737928428058450
