| L(s) = 1 | + (−1.32 − 2.69i)3-s − 0.640·5-s + 2.72·7-s + (−5.47 + 7.14i)9-s + 11.2·11-s − 5.25i·13-s + (0.849 + 1.72i)15-s − 14.8i·17-s − 15.0i·19-s + (−3.61 − 7.31i)21-s + 36.4i·23-s − 24.5·25-s + (26.4 + 5.24i)27-s + 51.7·29-s + 36.5·31-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.896i)3-s − 0.128·5-s + 0.388·7-s + (−0.608 + 0.793i)9-s + 1.02·11-s − 0.403i·13-s + (0.0566 + 0.114i)15-s − 0.874i·17-s − 0.793i·19-s + (−0.171 − 0.348i)21-s + 1.58i·23-s − 0.983·25-s + (0.980 + 0.194i)27-s + 1.78·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.435647127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435647127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.32 + 2.69i)T \) |
| good | 5 | \( 1 + 0.640T + 25T^{2} \) |
| 7 | \( 1 - 2.72T + 49T^{2} \) |
| 11 | \( 1 - 11.2T + 121T^{2} \) |
| 13 | \( 1 + 5.25iT - 169T^{2} \) |
| 17 | \( 1 + 14.8iT - 289T^{2} \) |
| 19 | \( 1 + 15.0iT - 361T^{2} \) |
| 23 | \( 1 - 36.4iT - 529T^{2} \) |
| 29 | \( 1 - 51.7T + 841T^{2} \) |
| 31 | \( 1 - 36.5T + 961T^{2} \) |
| 37 | \( 1 + 63.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 59.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 58.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 23.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 7.29iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 32.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 112. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80.0T + 9.40e3T^{2} \) |
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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
−9.852405299429053132663490838438, −8.920983221991124240256481783812, −7.984270962992785669571117504953, −7.24031578053648764776370478616, −6.44043994193663565786588189069, −5.48939471538011395379790198849, −4.54136751373462444155320177363, −3.12456418261152959767153947386, −1.80194634552905201400799259707, −0.58941572965562794571843378193,
1.28038868023937209792498328264, 3.01277270413070756230703794895, 4.26481160624391588625502426673, 4.65201550534156364203754131928, 6.18143845861341790159645047428, 6.46098695516595560820091529781, 8.111405916164456109891692051194, 8.634841785229259181708021437751, 9.749727097592457855734980024730, 10.27786982634467146894654516083
