| L(s) = 1 | − 3.27i·2-s + 1.25·3-s − 6.73·4-s − 4.10i·6-s − 3.94i·7-s + 8.97i·8-s − 7.43·9-s + (−6.83 − 8.61i)11-s − 8.43·12-s + 6.61i·13-s − 12.9·14-s + 2.44·16-s + 1.49i·17-s + 24.3i·18-s − 18.5i·19-s + ⋯ |
| L(s) = 1 | − 1.63i·2-s + 0.417·3-s − 1.68·4-s − 0.683i·6-s − 0.563i·7-s + 1.12i·8-s − 0.825·9-s + (−0.621 − 0.783i)11-s − 0.702·12-s + 0.509i·13-s − 0.922·14-s + 0.152·16-s + 0.0876i·17-s + 1.35i·18-s − 0.977i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.422564 + 0.874256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422564 + 0.874256i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (6.83 + 8.61i)T \) |
| good | 2 | \( 1 + 3.27iT - 4T^{2} \) |
| 3 | \( 1 - 1.25T + 9T^{2} \) |
| 7 | \( 1 + 3.94iT - 49T^{2} \) |
| 13 | \( 1 - 6.61iT - 169T^{2} \) |
| 17 | \( 1 - 1.49iT - 289T^{2} \) |
| 19 | \( 1 + 18.5iT - 361T^{2} \) |
| 23 | \( 1 + 18.1T + 529T^{2} \) |
| 29 | \( 1 - 24.3iT - 841T^{2} \) |
| 31 | \( 1 + 16.8T + 961T^{2} \) |
| 37 | \( 1 - 65.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 81.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 75.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 4.85iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 12.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 79.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 42.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 77.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 37.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 102.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 11.4T + 9.40e3T^{2} \) |
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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
−10.99082740704039805199918422673, −10.47307151149670235704723435653, −9.231762646294220614293608845770, −8.658258823274776792515937173070, −7.34671610484139465521544276839, −5.69593452348293353849142515576, −4.27551124669365425147960231908, −3.23758794246787053125275189064, −2.21951525396654573734977351953, −0.44441382399085907021076945638,
2.60066219091711568433175871640, 4.37144057483473114438148440350, 5.61108919661833256624161374515, 6.17786295407700550637458378877, 7.73126184481126929823326326841, 7.983582736142862002611663121218, 9.092467930171954027727153063905, 9.945580601480142014399239231189, 11.40042834919988642431554595516, 12.56946386197831728650374857044
