| L(s) = 1 | − 15.5i·3-s − 150·5-s − 325. i·7-s − 243·9-s − 1.47e3i·11-s − 3.39e3·13-s + 2.33e3i·15-s + 5.17e3·17-s − 6.81e3i·19-s − 5.07e3·21-s − 3.99e3i·23-s + 6.87e3·25-s + 3.78e3i·27-s − 3.21e4·29-s + 3.26e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.19·5-s − 0.949i·7-s − 0.333·9-s − 1.10i·11-s − 1.54·13-s + 0.692i·15-s + 1.05·17-s − 0.992i·19-s − 0.548·21-s − 0.327i·23-s + 0.440·25-s + 0.192i·27-s − 1.31·29-s + 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.003092525294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003092525294\) |
| \(L(4)\) |
|
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 | \( 1 + 150T + 1.56e4T^{2} \) |
| 7 | \( 1 + 325. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.47e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.39e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 5.17e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 6.81e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.99e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.21e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.26e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.61e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 7.00e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.00e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.51e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 6.69e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 3.90e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.57e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.21e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.43e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.43e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.74e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.03e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.86e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 9.42e5T + 8.32e11T^{2} \) |
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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
−10.96287907639051323708870438404, −9.753768232606397729234863922061, −8.362207430228901083335386668424, −7.56166096547342427925018623594, −6.88596438807533889093712544667, −5.28927207194301954959714402871, −3.99433832887367642840355147350, −2.86554668320248084744490069978, −0.911667414241835449930565762604, −0.00108339349063662211504236978,
2.18095745078240950625752512643, 3.57409448465980891543459708572, 4.66241917898031283860126566046, 5.69652810973822116883021461183, 7.40586034193957804923174789471, 7.989416789333918966356810791783, 9.420794360761443201177363175831, 9.996998108111873356103877594881, 11.44075755328147395947945165349, 12.08644960372799131905528015474
