L(s) = 1 | + (2.06 − 2.17i)3-s + (−3.17 + 3.17i)5-s − 6.03i·7-s + (−0.485 − 8.98i)9-s + (13.0 − 13.0i)11-s + (6.39 − 6.39i)13-s + (0.363 + 13.4i)15-s − 4.39i·17-s + (3.21 − 3.21i)19-s + (−13.1 − 12.4i)21-s − 34.0·23-s + 4.78i·25-s + (−20.5 − 17.4i)27-s + (27.9 + 27.9i)29-s + 7.90·31-s + ⋯ |
L(s) = 1 | + (0.687 − 0.725i)3-s + (−0.635 + 0.635i)5-s − 0.862i·7-s + (−0.0539 − 0.998i)9-s + (1.18 − 1.18i)11-s + (0.491 − 0.491i)13-s + (0.0242 + 0.898i)15-s − 0.258i·17-s + (0.168 − 0.168i)19-s + (−0.626 − 0.593i)21-s − 1.47·23-s + 0.191i·25-s + (−0.761 − 0.647i)27-s + (0.964 + 0.964i)29-s + 0.255·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34418 - 1.05016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34418 - 1.05016i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.06 + 2.17i)T \) |
good | 5 | \( 1 + (3.17 - 3.17i)T - 25iT^{2} \) |
| 7 | \( 1 + 6.03iT - 49T^{2} \) |
| 11 | \( 1 + (-13.0 + 13.0i)T - 121iT^{2} \) |
| 13 | \( 1 + (-6.39 + 6.39i)T - 169iT^{2} \) |
| 17 | \( 1 + 4.39iT - 289T^{2} \) |
| 19 | \( 1 + (-3.21 + 3.21i)T - 361iT^{2} \) |
| 23 | \( 1 + 34.0T + 529T^{2} \) |
| 29 | \( 1 + (-27.9 - 27.9i)T + 841iT^{2} \) |
| 31 | \( 1 - 7.90T + 961T^{2} \) |
| 37 | \( 1 + (-20.0 - 20.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 45.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.0 - 36.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 5.08iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.7 + 20.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-39.0 + 39.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (49.8 - 49.8i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (44.9 - 44.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 46.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 97.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 40.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-35.5 - 35.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 69.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−11.99617182319561759760444529554, −11.29912793924269218016606254994, −10.17162114295177345779259141687, −8.834036366962198727808909144842, −7.985252325735722589678691383567, −7.01552463016341534156626847017, −6.15151485530384695854793605126, −3.94956141352901184667154470456, −3.13790267172169787183576892328, −1.02302796238893051162044707297,
2.06343582623498009513614868521, 3.89240347182650760159623264704, 4.60579362165841957657683801562, 6.15885269995043770413890596641, 7.72472726397183052151246693126, 8.670621760412539353155767093946, 9.350366403976212280094636644240, 10.33427974916937607574073142890, 11.90132813884751062592713314593, 12.14409557005039993323864771754