L(s) = 1 | + 2-s + (1.72 − 0.158i)3-s + 4-s + (−0.724 + 1.25i)5-s + (1.72 − 0.158i)6-s + 8-s + (2.94 − 0.548i)9-s + (−0.724 + 1.25i)10-s + (−1 − 1.73i)11-s + (1.72 − 0.158i)12-s + (2.44 + 4.24i)13-s + (−1.05 + 2.28i)15-s + 16-s + (1 − 1.73i)17-s + (2.94 − 0.548i)18-s + (1.27 + 2.20i)19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.995 − 0.0917i)3-s + 0.5·4-s + (−0.324 + 0.561i)5-s + (0.704 − 0.0648i)6-s + 0.353·8-s + (0.983 − 0.182i)9-s + (−0.229 + 0.396i)10-s + (−0.301 − 0.522i)11-s + (0.497 − 0.0458i)12-s + (0.679 + 1.17i)13-s + (−0.271 + 0.588i)15-s + 0.250·16-s + (0.242 − 0.420i)17-s + (0.695 − 0.129i)18-s + (0.292 + 0.506i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.28097 + 0.461234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.28097 + 0.461234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.72 + 0.158i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.724 - 1.25i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 - 2.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.44 + 5.97i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + (5.89 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 - 5.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-5.44 + 9.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + (3.44 - 5.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.44 + 2.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.18439378966835514890348239977, −9.282732002872879395957759001126, −8.403853634853568710424606175587, −7.52987215703642594901647504184, −6.83262111191519523933700334965, −5.88791524679931816619463832628, −4.57123502161400594395656962633, −3.65594481165534648506263846435, −2.94348281852844384207722159660, −1.69695934251557328543135873910,
1.43098205278338399079499359255, 2.86160507481221046413497990912, 3.63831447976259961942917756773, 4.68383518381932287360637471058, 5.44772261471286630358558427001, 6.77343874263190645450826834328, 7.62124289263485136274145945863, 8.393006692999381844047307276151, 9.062342318041978882532321046276, 10.28519180229230732329990025350