L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1 + 1.73i)5-s + (0.5 − 0.866i)7-s + 0.999·8-s − 1.99·10-s + (2 − 3.46i)11-s + (−3 − 5.19i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s − 4·19-s + (0.999 − 1.73i)20-s + (1.99 + 3.46i)22-s + (−4 − 6.92i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 + 0.774i)5-s + (0.188 − 0.327i)7-s + 0.353·8-s − 0.632·10-s + (0.603 − 1.04i)11-s + (−0.832 − 1.44i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.485·17-s − 0.917·19-s + (0.223 − 0.387i)20-s + (0.426 + 0.738i)22-s + (−0.834 − 1.44i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.158482485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158482485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.885778667519538311425585998505, −8.669388729820317307964743941555, −8.143009293789829425246102601709, −7.20578254342528921001303957977, −6.35931100083558245288351769371, −5.78327487432759317661103185469, −4.67956484641865475321130999341, −3.44220668104326879194820005947, −2.32742428023720118268291534390, −0.57246189713684143371210172076,
1.56740343359640761675872133826, 2.14861667714128077134085557197, 3.78792042335503094939221623224, 4.63157827510658929136200378650, 5.45594299134574925731759233087, 6.71890543664226226781911190105, 7.49471232616844452988196306763, 8.571211596348526381669538597836, 9.345475652651039134221630359466, 9.613628234917381576404022668893