| L(s) = 1 | − 11.3i·2-s − 128.·4-s − 503. i·5-s + 907.·7-s + 1.44e3i·8-s − 5.70e3·10-s + 5.07e3i·11-s + 2.73e4·13-s − 1.02e4i·14-s + 1.63e4·16-s − 1.52e5i·17-s + 6.06e4·19-s + 6.45e4i·20-s + 5.74e4·22-s + 2.41e5i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.806i·5-s + 0.377·7-s + 0.353i·8-s − 0.570·10-s + 0.346i·11-s + 0.957·13-s − 0.267i·14-s + 0.250·16-s − 1.82i·17-s + 0.465·19-s + 0.403i·20-s + 0.245·22-s + 0.863i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.871667698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.871667698\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 11.3iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
| good | 5 | \( 1 + 503. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 5.07e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.73e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.52e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 6.06e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.41e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 9.45e3iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.56e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.40e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.90e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 9.49e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.22e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.87e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.66e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.42e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.24e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.34e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.10e5T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.16e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.26e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 9.95e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.58e7T + 7.83e15T^{2} \) |
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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
−11.61513881011085868736991868594, −10.40296027748278177285985777260, −9.289187160237454896310609155412, −8.504553138107560825790291009453, −7.18448449027848083314134799364, −5.43816341302605632649899193549, −4.55364794826743291996662843175, −3.13643858111705415550982060237, −1.60072535291302291888022687067, −0.55241251523581725490336583090,
1.29111888180504623000995342816, 3.10704787842626926485743441646, 4.38094914706880345317419381814, 5.92051151618857161190374968314, 6.65241454408254354354334909542, 7.996092992540330130535952108882, 8.741097331543496722282739725376, 10.29002990639347164012375543838, 10.97299274820410573864701951078, 12.31932583999510113674761409856
