L(s) = 1 | + (3.19 − 1.84i)3-s + (2.63 + 1.52i)5-s + (0.812 + 6.95i)7-s + (2.28 − 3.96i)9-s + (1.17 + 2.03i)11-s + 25.3i·13-s + 11.2·15-s + (3.08 − 1.78i)17-s + (14.1 + 8.18i)19-s + (15.4 + 20.6i)21-s + (8.83 − 15.3i)23-s + (−7.85 − 13.6i)25-s + 16.2i·27-s − 36.1·29-s + (6.25 − 3.61i)31-s + ⋯ |
L(s) = 1 | + (1.06 − 0.614i)3-s + (0.527 + 0.304i)5-s + (0.116 + 0.993i)7-s + (0.254 − 0.440i)9-s + (0.106 + 0.185i)11-s + 1.94i·13-s + 0.748·15-s + (0.181 − 0.104i)17-s + (0.746 + 0.430i)19-s + (0.733 + 0.985i)21-s + (0.384 − 0.665i)23-s + (−0.314 − 0.544i)25-s + 0.603i·27-s − 1.24·29-s + (0.201 − 0.116i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.691522109\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.691522109\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.812 - 6.95i)T \) |
good | 3 | \( 1 + (-3.19 + 1.84i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-2.63 - 1.52i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 2.03i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 25.3iT - 169T^{2} \) |
| 17 | \( 1 + (-3.08 + 1.78i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-14.1 - 8.18i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-8.83 + 15.3i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 36.1T + 841T^{2} \) |
| 31 | \( 1 + (-6.25 + 3.61i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-18.4 + 31.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 53.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (27.1 + 15.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (35.1 + 60.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-81.4 + 47.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-1.89 - 1.09i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-12.4 - 21.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (68.9 - 39.7i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-57.5 + 99.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 154. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-98.7 - 57.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 53.9iT - 9.40e3T^{2} \) |
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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
−11.08923930635096199613704529437, −9.638307501979673618266098750181, −9.172859924801108087832951536555, −8.326958493882966384778372180708, −7.30677618523229416685410306789, −6.45399794112752892328679503855, −5.33179556091908976329173766776, −3.85414333836155460471153145404, −2.43120477615304986341673176902, −1.87074318743060312254976197219,
1.07684681945363863512109566371, 2.91213920178664782974234006545, 3.65784830331499686688340797359, 4.93494787183548746210706094716, 5.94597478094911139262833050105, 7.51285486814083521491362059638, 8.034760020852551499406481445042, 9.194873416068035517152575115296, 9.783271039252336160107352008078, 10.55297869175768604167120415279