| L(s) = 1 | + 46.7i·3-s + 26.4·5-s − 1.82e3i·7-s − 2.18e3·9-s − 877. i·11-s + 1.33e4·13-s + 1.23e3i·15-s + 1.42e5·17-s + 4.79e4i·19-s + 8.54e4·21-s + 9.51e4i·23-s − 3.89e5·25-s − 1.02e5i·27-s − 1.23e6·29-s + 8.41e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.0423·5-s − 0.761i·7-s − 0.333·9-s − 0.0599i·11-s + 0.465·13-s + 0.0244i·15-s + 1.70·17-s + 0.367i·19-s + 0.439·21-s + 0.340i·23-s − 0.998·25-s − 0.192i·27-s − 1.75·29-s + 0.911i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.212462197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212462197\) |
| \(L(5)\) |
|
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 | \( 1 - 26.4T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.82e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 877. iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.33e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.42e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 4.79e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 9.51e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.23e6T + 5.00e11T^{2} \) |
| 31 | \( 1 - 8.41e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.47e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.07e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.31e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.04e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 2.11e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.08e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 3.50e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.35e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 5.14e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 6.92e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 4.83e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.30e6T + 3.93e15T^{2} \) |
| 97 | \( 1 - 9.38e7T + 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.997857871210927310178388959975, −9.266881299816096208226613464278, −8.051263400297218004442713826664, −7.35647682586225714992804880276, −6.01815244130582340779531956577, −5.23014544125172578056063703097, −3.91573172159054674605048088072, −3.36816956014662733637503020319, −1.75523340888457105948642966441, −0.60528513210147075012343935176,
0.72950810485812357872136621699, 1.83758237071040354134730805146, 2.86492576031581561271925741796, 4.03434302763717846768589085936, 5.56812016066011633705051104523, 5.97654066400910781045715872693, 7.34686631297219540166371164150, 8.005313530786612124111379980224, 9.072153250616075043624365661552, 9.835965094582222304674885893547
