| L(s) = 1 | + 0.850i·2-s − 5.19·3-s + 15.2·4-s − 11.2·5-s − 4.41i·6-s − 26.4·7-s + 26.5i·8-s + 27·9-s − 9.58i·10-s + 19.9i·11-s − 79.3·12-s − 190. i·13-s − 22.4i·14-s + 58.5·15-s + 221.·16-s − 159.·17-s + ⋯ |
| L(s) = 1 | + 0.212i·2-s − 0.577·3-s + 0.954·4-s − 0.450·5-s − 0.122i·6-s − 0.539·7-s + 0.415i·8-s + 0.333·9-s − 0.0958i·10-s + 0.165i·11-s − 0.551·12-s − 1.12i·13-s − 0.114i·14-s + 0.260·15-s + 0.866·16-s − 0.551·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7817990821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7817990821\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (1.20e3 - 3.26e3i)T \) |
| good | 2 | \( 1 - 0.850iT - 16T^{2} \) |
| 5 | \( 1 + 11.2T + 625T^{2} \) |
| 7 | \( 1 + 26.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 19.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 190. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 159.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 294.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 165. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 513.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.62e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.77e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.06e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.18e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 246.T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.49e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 3.88e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.86e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 619. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.37e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.60e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.91e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.37e3iT - 8.85e7T^{2} \) |
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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
−11.71406291209168216648691258267, −10.77281558014440253233379893234, −10.01583322190417147929103283905, −8.413180284227111881655125835539, −7.37222005076413838360990828173, −6.43140794175943665057214102228, −5.47617112410286644519821037208, −3.84240430406884883214945838288, −2.29957382935338180604031963879, −0.29071644494496947263790160418,
1.61964668909331041654599999937, 3.22548575748504191903866118948, 4.61878314838349339448227407046, 6.31340256057755032067247393472, 6.78191967199364188055043599923, 8.112337048168993951058930851795, 9.512574809194706727708100997786, 10.57956572355885774744047062725, 11.40832370188374128900655135983, 12.07307201785154505093644119570
