L(s) = 1 | + i·2-s − 4-s − i·8-s + 11-s + 16-s − i·17-s + 19-s + i·22-s + i·32-s + 34-s + i·38-s + 41-s + 2i·43-s − 44-s − 49-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·8-s + 11-s + 16-s − i·17-s + 19-s + i·22-s + i·32-s + 34-s + i·38-s + 41-s + 2i·43-s − 44-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.097653936\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097653936\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - 2iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−9.522427778975034796057817967191, −8.775576639026423808114104785420, −7.908410925959599197103503834839, −7.18795282510836903390806698690, −6.51105910546158158543828620353, −5.66682520410133423496118719312, −4.83169195739333518555481908427, −3.99185489963493349396478221504, −2.95654449017666676530158671064, −1.14818931593202013345070836726,
1.17736854813761512151093735038, 2.25594644678019811115267924317, 3.48866509072144653531176499563, 4.04969398771955654931045658331, 5.12558399548621274911847246335, 5.95043427263604448866412673948, 6.98562482909292943684520635666, 7.997213166894221633581959269714, 8.758361273880077137412131551180, 9.423014382568503879193719248955