| L(s) = 1 | − 168. i·5-s − 180.·7-s − 485. i·11-s − 3.11e3·13-s − 2.71e3i·17-s + 4.42e3·19-s + 2.32e4i·23-s − 1.29e4·25-s + 2.71e3i·29-s + 5.19e4·31-s + 3.05e4i·35-s − 8.12e4·37-s − 1.72e3i·41-s − 1.49e5·43-s + 1.32e5i·47-s + ⋯ |
| L(s) = 1 | − 1.35i·5-s − 0.527·7-s − 0.364i·11-s − 1.41·13-s − 0.553i·17-s + 0.644·19-s + 1.90i·23-s − 0.827·25-s + 0.111i·29-s + 1.74·31-s + 0.713i·35-s − 1.60·37-s − 0.0250i·41-s − 1.87·43-s + 1.27i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1179671757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1179671757\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 168. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 180.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 485. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.11e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.71e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.42e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.32e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.71e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.19e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 8.12e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.72e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.49e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.32e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.16e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 9.87e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.97e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.99e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.88e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.33e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.22e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.89e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.88e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.00e6T + 8.32e11T^{2} \) |
| 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
−12.26977489042562751666231152085, −11.63926490871987166845179150003, −9.943988127636762784637781726745, −9.345805703315398899511121321095, −8.206165871569265100181088029057, −7.07879488290378762785880692062, −5.52056758036774054107592275116, −4.69709187993958389202211604786, −3.09571649482216793809690434449, −1.31685399162261845779839260449,
0.03596216560502023508103282288, 2.28429776018558872723161834976, 3.29456524525618020768179832528, 4.86876708638747858428487879980, 6.49911361000375296872837941746, 7.05569250140518716578774028023, 8.381051070561494612029480947808, 10.02472055850991110203764466484, 10.26001898751284879563476178821, 11.66839684604679021674843908745
