| L(s) = 1 | − 15.5i·3-s − 90·5-s + 187. i·7-s − 243·9-s + 1.68e3i·11-s + 1.76e3·13-s + 1.40e3i·15-s − 1.63e3·17-s + 1.25e4i·19-s + 2.91e3·21-s + 1.34e4i·23-s − 7.52e3·25-s + 3.78e3i·27-s − 1.60e4·29-s − 1.59e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.719·5-s + 0.545i·7-s − 0.333·9-s + 1.26i·11-s + 0.802·13-s + 0.415i·15-s − 0.333·17-s + 1.82i·19-s + 0.314·21-s + 1.10i·23-s − 0.481·25-s + 0.192i·27-s − 0.656·29-s − 0.533i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.704045 + 0.704045i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.704045 + 0.704045i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5iT \) |
| good | 5 | \( 1 + 90T + 1.56e4T^{2} \) |
| 7 | \( 1 - 187. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.68e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.76e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.63e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.25e4iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.34e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.60e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.59e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.11e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 9.85e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.91e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.78e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.75e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.54e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.06e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 5.51e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 7.40e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.00e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.72e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.44e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 8.19e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 5.57e5T + 8.32e11T^{2} \) |
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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
−14.83033499819214499371866945940, −13.40006072079276898711954813755, −12.27890115703722200891691289774, −11.49347348744092181672623014580, −9.860498255211320020148766604572, −8.350490584328310373706861894247, −7.30124885818590674597486299946, −5.75694010643262529347498383508, −3.85481791000258144551598889616, −1.79451457631363740340144601986,
0.45377696065171287955806151943, 3.26109890373613600549206703152, 4.58864513081078476443650376440, 6.39588933859475429158042343589, 8.047772789195789620607621118087, 9.159433852017253721744588158158, 10.85026266452845544420297383906, 11.36230219555731968377146946237, 13.10159845631660759744040501080, 14.13198979256058417946678427651
