L(s) = 1 | + 2i·3-s + 7-s − 9-s − 4i·11-s + 4i·13-s − 2·17-s + 6i·19-s + 2i·21-s − 8·23-s + 5·25-s + 4i·27-s − 2i·29-s − 4·31-s + 8·33-s + 10i·37-s + ⋯ |
L(s) = 1 | + 1.15i·3-s + 0.377·7-s − 0.333·9-s − 1.20i·11-s + 1.10i·13-s − 0.485·17-s + 1.37i·19-s + 0.436i·21-s − 1.66·23-s + 25-s + 0.769i·27-s − 0.371i·29-s − 0.718·31-s + 1.39·33-s + 1.64i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.452839816\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.452839816\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.672732593597194950984025557283, −8.785723803933609650617371643745, −8.291998783204522234294349930134, −7.23902651652528035790225865926, −6.14704461923121500029274219723, −5.54633344390425667840264944717, −4.30876348263583370054341056486, −4.06853526969133775380481934010, −2.89069247004404218362813725589, −1.50414249428293699898402646599,
0.54339113468104978389815767993, 1.88291078226632566944676563473, 2.56579511387070637701763941501, 4.02943360239990204590868483715, 4.94178959045265490855925218119, 5.87072247714935711127050846045, 6.82433041023246349280288512703, 7.39154135809763517554922142448, 7.946775133717328822803951042244, 8.886822846202992888782878355008