L(s) = 1 | − 168.·5-s + 13.9·7-s + 1.50e3·11-s + 744. i·13-s + 2.96e3i·17-s − 6.31e3i·19-s − 9.99e3i·23-s + 1.29e4·25-s − 2.03e4·29-s − 3.92e4·31-s − 2.36e3·35-s + 8.92e4i·37-s − 2.00e4i·41-s + 5.54e4i·43-s − 1.08e5i·47-s + ⋯ |
L(s) = 1 | − 1.35·5-s + 0.0407·7-s + 1.12·11-s + 0.339i·13-s + 0.603i·17-s − 0.920i·19-s − 0.821i·23-s + 0.827·25-s − 0.836·29-s − 1.31·31-s − 0.0551·35-s + 1.76i·37-s − 0.291i·41-s + 0.697i·43-s − 1.04i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.292598789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292598789\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 168.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 13.9T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.50e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 744. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 2.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.31e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 9.99e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.03e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.92e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.92e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 2.00e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.54e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.08e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.43e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.10e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.20e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.03e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.39e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.79e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.85e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.10e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.99e5T + 8.32e11T^{2} \) |
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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
−9.611628173463330788849125555201, −8.775619248826997363396196709561, −8.013801728814157761660855271814, −7.06204983354864707760727985283, −6.34305495590280837514541060015, −4.88852120917316178667347763602, −4.04793634995020450006010715352, −3.28347100237984060862696454452, −1.75263093218405809362981184938, −0.48729923973180615265855798306,
0.55304504019892342790653167504, 1.81019936004432576559186091656, 3.48182117065188918052104912569, 3.88148108095147378544312463887, 5.10651455429078270220755618879, 6.22073752457219897059155608277, 7.39684804291689991979344699542, 7.79656547692102872058519311380, 8.946832281957511182145056864753, 9.623190341560673953441185509355