| L(s) = 1 | − 46.7i·3-s − 726·5-s + 3.05e3i·7-s − 2.18e3·9-s + 1.32e4i·11-s − 3.90e4·13-s + 3.39e4i·15-s − 6.58e4·17-s − 1.30e5i·19-s + 1.42e5·21-s + 5.02e5i·23-s + 1.36e5·25-s + 1.02e5i·27-s − 2.02e5·29-s + 1.19e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.16·5-s + 1.27i·7-s − 0.333·9-s + 0.907i·11-s − 1.36·13-s + 0.670i·15-s − 0.787·17-s − 0.999i·19-s + 0.734·21-s + 1.79i·23-s + 0.349·25-s + 0.192i·27-s − 0.285·29-s + 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3721663323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3721663323\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 + 726T + 3.90e5T^{2} \) |
| 7 | \( 1 - 3.05e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.32e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.90e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 6.58e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.30e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 5.02e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 2.02e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.19e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.87e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.09e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.26e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.35e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.06e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.76e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.71e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.74e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.98e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.32e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.82e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 7.78e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.66e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 7.39e7T + 7.83e15T^{2} \) |
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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
−11.17513710424716372152590745335, −9.605679495620223711953035451261, −8.765986195443750263142854088507, −7.58594207577390372571239401753, −7.01299975256885672815583259933, −5.49429930017788776955338228201, −4.46925230223090905293057131511, −2.91902204570390844733044676306, −1.91507540596991583429384839084, −0.13454411129309862285288249384,
0.66558703464493010180718701347, 2.73517508324664461255625713971, 4.06683794083073691805518643085, 4.48370642596426067518237760931, 6.17092911418640255175017106012, 7.48489104635473035342332162386, 8.097004784722189091895479379890, 9.407756429300799375277238845743, 10.50132535522311251840591056381, 11.13402362399870035272037751365
