| L(s) = 1 | + 21.9i·3-s − 1.06e3i·7-s + 1.70e3·9-s − 671.·11-s + 1.41e3i·13-s − 2.41e4i·17-s + 6.79e3·19-s + 2.34e4·21-s + 4.10e4i·23-s + 8.55e4i·27-s − 1.74e5·29-s − 1.76e5·31-s − 1.47e4i·33-s − 6.99e4i·37-s − 3.10e4·39-s + ⋯ |
| L(s) = 1 | + 0.470i·3-s − 1.17i·7-s + 0.778·9-s − 0.152·11-s + 0.178i·13-s − 1.19i·17-s + 0.227·19-s + 0.553·21-s + 0.704i·23-s + 0.836i·27-s − 1.32·29-s − 1.06·31-s − 0.0714i·33-s − 0.226i·37-s − 0.0837·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.200398187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200398187\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 21.9iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 1.06e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 671.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.41e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.41e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 6.79e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.10e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.74e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.76e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.99e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.61e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.63e3iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.44e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.88e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.80e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.24e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.54e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.53e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 6.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.43e7iT - 8.07e13T^{2} \) |
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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
−10.76178144603093635952460522637, −9.906346141051305095679902802741, −9.131535701570664702783902246493, −7.52512979788372904008843605746, −7.06141275008525784831830280287, −5.41707368983850629688019835697, −4.32805067835948789174106668385, −3.41520608466481865139603922181, −1.64248613507633493948386562375, −0.29365801325341427524108373942,
1.41251318166437453649905010466, 2.45177922699501671464817991220, 3.95193800203210454093198176705, 5.37778503287338693668657665877, 6.31506624164342156854944591596, 7.48263800648040999139476774786, 8.466642745549849083472374842712, 9.450259800296087191904900084510, 10.53640424143370803889691468261, 11.61555326484966706940413351164
