| L(s) = 1 | + 49.7i·3-s − 570.·5-s − 2.53e3i·7-s + 4.08e3·9-s − 1.94e3i·11-s + 1.22e4·13-s − 2.83e4i·15-s + 1.34e5·17-s − 1.51e5i·19-s + 1.26e5·21-s − 4.94e5i·23-s − 6.46e4·25-s + 5.29e5i·27-s + 7.87e5·29-s − 7.54e5i·31-s + ⋯ |
| L(s) = 1 | + 0.614i·3-s − 0.913·5-s − 1.05i·7-s + 0.622·9-s − 0.132i·11-s + 0.429·13-s − 0.560i·15-s + 1.60·17-s − 1.15i·19-s + 0.648·21-s − 1.76i·23-s − 0.165·25-s + 0.996i·27-s + 1.11·29-s − 0.816i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.35837 - 0.562657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35837 - 0.562657i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 49.7iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 570.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 2.53e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.94e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.22e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.34e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.51e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.94e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.87e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 7.54e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.61e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.32e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.74e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.28e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.94e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.43e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.09e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.96e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 6.82e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.92e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.48e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.84e5iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.79e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.34e8T + 7.83e15T^{2} \) |
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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
−15.03646486544657112183379377684, −13.73027394471955025486049617661, −12.30840507759929857747445972650, −10.90926842048715327092763916256, −9.954881058154778850356504709933, −8.208577023606869142340009680713, −6.92322478935387533210704729701, −4.66425744976994601803639143445, −3.54996924102801156457572812653, −0.71451572553120339128802928210,
1.45720793078430547528771700103, 3.55653833174547504935958443367, 5.59878434145335593576890665163, 7.31101212905223389738315683109, 8.381442377862705135674973255293, 10.06198448724586013931703254670, 11.92278340722344893762519451127, 12.32923747625861611799754001879, 13.91260848197739147719861892463, 15.34574916059911911941830493805
