| L(s) = 1 | + 46.7i·3-s + 1.11e3·5-s + 1.26e3i·7-s − 2.18e3·9-s + 2.17e4i·11-s + 3.95e4·13-s + 5.22e4i·15-s + 1.35e5·17-s + 1.16e5i·19-s − 5.92e4·21-s − 6.81e4i·23-s + 8.59e5·25-s − 1.02e5i·27-s − 4.40e5·29-s + 4.20e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.78·5-s + 0.527i·7-s − 0.333·9-s + 1.48i·11-s + 1.38·13-s + 1.03i·15-s + 1.61·17-s + 0.897i·19-s − 0.304·21-s − 0.243i·23-s + 2.20·25-s − 0.192i·27-s − 0.622·29-s + 0.0455i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.137034405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.137034405\) |
| \(L(5)\) |
|
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 | \( 1 - 1.11e3T + 3.90e5T^{2} \) |
| 7 | \( 1 - 1.26e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.17e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.95e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.35e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.16e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 6.81e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.40e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 4.20e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.37e4T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.67e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.32e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.47e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.57e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.45e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.17e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.79e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.38e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.56e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.26e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 7.23e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.15e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 7.87e7T + 7.83e15T^{2} \) |
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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
−10.06301657595652549103468480857, −9.494021165591650098436377718576, −8.690938461691420614194567010474, −7.39077083727986693619203609089, −5.91482722401554637029549675935, −5.77964320737180522706299183113, −4.52703305625543671709879157104, −3.20635988634513673858737527821, −2.03082517998519812930197163229, −1.28559641632455656684640451411,
0.857988184608364282703834287739, 1.31149277457227428414223963467, 2.62813964142535218947074474134, 3.60540719683553363379403482405, 5.42297990592172498348746595973, 5.90554132127323718044416865191, 6.70304292923944839821075206784, 7.958541376576090739207198000520, 8.898567610583996455032446890197, 9.695756081976977266450038513479
