L(s) = 1 | + (−0.866 + 1.5i)3-s + (0.5 − 0.866i)5-s + (−1.86 − 3.23i)7-s + (−1.5 − 2.59i)9-s + (−2.73 − 4.73i)11-s + (−0.732 + 1.26i)13-s + (0.866 + 1.5i)15-s + 7.46·17-s − 2·19-s + 6.46·21-s + (−0.133 + 0.232i)23-s + (−0.499 − 0.866i)25-s + 5.19·27-s + (−4.23 − 7.33i)29-s + (1 − 1.73i)31-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (0.223 − 0.387i)5-s + (−0.705 − 1.22i)7-s + (−0.5 − 0.866i)9-s + (−0.823 − 1.42i)11-s + (−0.203 + 0.351i)13-s + (0.223 + 0.387i)15-s + 1.81·17-s − 0.458·19-s + 1.41·21-s + (−0.0279 + 0.0483i)23-s + (−0.0999 − 0.173i)25-s + 1.00·27-s + (−0.785 − 1.36i)29-s + (0.179 − 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.611509 - 0.513117i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611509 - 0.513117i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.86 + 3.23i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.732 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (0.133 - 0.232i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.23 + 7.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + (-1.96 + 3.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.73 - 9.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.86 + 3.23i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-3.19 + 5.53i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.767 - 1.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.86 - 8.42i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + (4.26 + 7.39i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.40 - 2.42i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + (2.46 + 4.26i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.04556317398734608612697142215, −10.24022340520916970478173356885, −9.728220171936271206516373528019, −8.556095309332620402026639552942, −7.47103326138563014872651638181, −6.12528411835334544016819336051, −5.41967964398944480144057992240, −4.09945927414284025515760451864, −3.21904497362344097652353402901, −0.56731375051423729528979153887,
1.98143224667969022500237231600, 3.06226312921889317955751392042, 5.20002225972975219702325681800, 5.75293896292793653187458719451, 6.95366829851945055508569903432, 7.63688865967392468488734496618, 8.835205318974540724487739788257, 9.989345208533836676916113642185, 10.63149979767634279884820600630, 12.07629423352292187683560154135