L(s) = 1 | + (0.195 − 0.980i)2-s + (0.555 + 0.831i)3-s + (−0.923 − 0.382i)4-s + (0.980 + 0.195i)5-s + (0.923 − 0.382i)6-s + (−0.555 + 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (−0.195 − 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (0.831 − 0.555i)17-s + (0.831 + 0.555i)18-s + (−0.707 + 0.292i)19-s + (−0.831 − 0.555i)20-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)2-s + (0.555 + 0.831i)3-s + (−0.923 − 0.382i)4-s + (0.980 + 0.195i)5-s + (0.923 − 0.382i)6-s + (−0.555 + 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (−0.195 − 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (0.831 − 0.555i)17-s + (0.831 + 0.555i)18-s + (−0.707 + 0.292i)19-s + (−0.831 − 0.555i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.388182853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388182853\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.195 + 0.980i)T \) |
| 3 | \( 1 + (-0.555 - 0.831i)T \) |
| 5 | \( 1 + (-0.980 - 0.195i)T \) |
| 17 | \( 1 + (-0.831 + 0.555i)T \) |
good | 7 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.785 + 1.17i)T + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.275 + 0.275i)T - iT^{2} \) |
| 53 | \( 1 + (0.425 + 1.02i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.636 + 1.53i)T + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.923 + 0.382i)T^{2} \) |
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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
−10.27707144606515083311687315278, −9.345924837879335778593197747833, −8.941668557711726037839346610871, −7.947101433557445275667309166878, −6.53703085141967840262231928097, −5.37566535666650858531798592417, −4.83685152544946425007665127446, −3.61187847510969846436977109108, −2.81110309351978217735781622570, −1.79193683910883543721680266992,
1.48671161032624166855222918889, 2.90822907898284402434470845519, 4.04592215888554030020983844566, 5.43040288340459782995843249522, 5.94825308227774981843472010041, 6.87940727524130098827890033058, 7.58770222344272561210522615796, 8.432918353148709480757091672285, 9.215983266351575363614338376842, 9.710895749996658217892586549048