| L(s) = 1 | + 4.88i·3-s + 2.32·5-s − 14.8·9-s − 15.3i·11-s − 14.8·13-s + 11.3i·15-s + 2.39·17-s + 29.9i·19-s − 26.5i·23-s − 19.6·25-s − 28.6i·27-s − 4.75·29-s − 19.7i·31-s + 74.8·33-s + 48.7·37-s + ⋯ |
| L(s) = 1 | + 1.62i·3-s + 0.464·5-s − 1.65·9-s − 1.39i·11-s − 1.14·13-s + 0.755i·15-s + 0.141·17-s + 1.57i·19-s − 1.15i·23-s − 0.784·25-s − 1.05i·27-s − 0.164·29-s − 0.636i·31-s + 2.26·33-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9422160128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9422160128\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4.88iT - 9T^{2} \) |
| 5 | \( 1 - 2.32T + 25T^{2} \) |
| 11 | \( 1 + 15.3iT - 121T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 - 2.39T + 289T^{2} \) |
| 19 | \( 1 - 29.9iT - 361T^{2} \) |
| 23 | \( 1 + 26.5iT - 529T^{2} \) |
| 29 | \( 1 + 4.75T + 841T^{2} \) |
| 31 | \( 1 + 19.7iT - 961T^{2} \) |
| 37 | \( 1 - 48.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 37.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 88.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 17.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 36.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 88.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 14.8T + 9.40e3T^{2} \) |
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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
−9.383061944680709229858435136389, −8.492158147939282977321056305033, −7.85217248695962194331348012945, −6.40780952602027142924779119863, −5.66674693715752065166155211496, −5.02080222390462907836811150209, −4.00433357856061170614471890027, −3.34121664186695758072185403195, −2.20546741915855906334739078503, −0.25455259576145993904200879399,
1.21093995130486070183687447628, 2.13405825790575222701300227909, 2.81754000355144506521211399300, 4.50520459015240422442881368514, 5.35911664663702330394173808212, 6.33038910813114143338760954900, 7.14436572164356010816282022746, 7.43277122236250102407386264769, 8.306935184856629801875412855724, 9.556952672960534643995306145107
