L(s) = 1 | + (0.707 − 0.707i)3-s − 5-s + 1.41i·7-s − 1.00i·9-s + (−0.707 + 0.707i)15-s + (1.00 + 1.00i)21-s + 1.41·23-s + 25-s + (−0.707 − 0.707i)27-s − 1.41i·35-s + 2i·41-s + 1.41·43-s + 1.00i·45-s + 1.41·47-s − 1.00·49-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s − 5-s + 1.41i·7-s − 1.00i·9-s + (−0.707 + 0.707i)15-s + (1.00 + 1.00i)21-s + 1.41·23-s + 25-s + (−0.707 − 0.707i)27-s − 1.41i·35-s + 2i·41-s + 1.41·43-s + 1.00i·45-s + 1.41·47-s − 1.00·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.391697130\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.391697130\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.639163022862277360086964484793, −7.996603816067804511440668886636, −7.31689532647479686569068978226, −6.60417902498538967695676089779, −5.78808989137020646873215763814, −4.88684051379152758422520559656, −3.90952483144489179310417362983, −2.96095573817742705856300078850, −2.46337935622586281392649732504, −1.12418347067374407920736451455,
0.907516935546230755221536131392, 2.47093531157134890589620554362, 3.48764252977294784069339095837, 3.96092236669099698664225630391, 4.61360348343072813955555176988, 5.42832202003097371252154260544, 6.80014998803149132875143788165, 7.40519660058103081992144492389, 7.80088564320645050970796679859, 8.774841428552050549519257302023