L(s) = 1 | − 2.82i·5-s − 3·7-s − 19.7i·11-s + 7·13-s − 14.1i·17-s − 19·19-s − 2.82i·23-s + 17·25-s − 50.9i·29-s − 10·31-s + 8.48i·35-s + 63·37-s + 50.9i·41-s − 50·43-s − 42.4i·47-s + ⋯ |
L(s) = 1 | − 0.565i·5-s − 0.428·7-s − 1.79i·11-s + 0.538·13-s − 0.831i·17-s − 19-s − 0.122i·23-s + 0.680·25-s − 1.75i·29-s − 0.322·31-s + 0.242i·35-s + 1.70·37-s + 1.24i·41-s − 1.16·43-s − 0.902i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.892200 - 0.892200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.892200 - 0.892200i\) |
\(L(2)\) |
|
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 | \( 1 + 2.82iT - 25T^{2} \) |
| 7 | \( 1 + 3T + 49T^{2} \) |
| 11 | \( 1 + 19.7iT - 121T^{2} \) |
| 13 | \( 1 - 7T + 169T^{2} \) |
| 17 | \( 1 + 14.1iT - 289T^{2} \) |
| 19 | \( 1 + 19T + 361T^{2} \) |
| 23 | \( 1 + 2.82iT - 529T^{2} \) |
| 29 | \( 1 + 50.9iT - 841T^{2} \) |
| 31 | \( 1 + 10T + 961T^{2} \) |
| 37 | \( 1 - 63T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50T + 1.84e3T^{2} \) |
| 47 | \( 1 + 42.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 73.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 98.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79T + 3.72e3T^{2} \) |
| 67 | \( 1 - 77T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 42.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 97T + 9.40e3T^{2} \) |
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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
−11.70977860610617227098567862651, −11.04506956382298680623223183801, −9.825943646897817915613913547795, −8.774299524182894108602763993439, −8.101326986853750701934163540746, −6.53007635444527891597178741168, −5.67992556246414970759367312034, −4.24801590395536063941728662376, −2.88492772039715598952188767432, −0.71148437845598326309032742967,
1.98666229873596286392538499627, 3.57282744992988254813882842688, 4.86084522784574180344487689848, 6.40110719362524553800153061383, 7.09492117274099749645533492537, 8.359929886661361973947746881358, 9.539435407952949826363109546005, 10.39482482914812763809836030204, 11.22638498827461634904094621148, 12.69305762787801526711545289047