L(s) = 1 | + (0.341 + 2.20i)5-s + (−2.45 + 4.25i)7-s + (1.62 − 2.81i)11-s + (−3.94 + 2.27i)13-s − 4.68·17-s + 2.29i·19-s + (2.41 − 1.39i)23-s + (−4.76 + 1.50i)25-s + (−0.841 − 0.485i)29-s + (8.33 − 4.81i)31-s + (−10.2 − 3.97i)35-s + 2.32i·37-s + (−8.23 + 4.75i)41-s + (0.256 − 0.443i)43-s + (−7.37 − 4.25i)47-s + ⋯ |
L(s) = 1 | + (0.152 + 0.988i)5-s + (−0.928 + 1.60i)7-s + (0.489 − 0.847i)11-s + (−1.09 + 0.631i)13-s − 1.13·17-s + 0.527i·19-s + (0.504 − 0.291i)23-s + (−0.953 + 0.301i)25-s + (−0.156 − 0.0902i)29-s + (1.49 − 0.864i)31-s + (−1.73 − 0.672i)35-s + 0.381i·37-s + (−1.28 + 0.742i)41-s + (0.0390 − 0.0676i)43-s + (−1.07 − 0.620i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4272434291\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4272434291\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.341 - 2.20i)T \) |
good | 7 | \( 1 + (2.45 - 4.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.62 + 2.81i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.94 - 2.27i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 + (-2.41 + 1.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.841 + 0.485i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.33 + 4.81i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.32iT - 37T^{2} \) |
| 41 | \( 1 + (8.23 - 4.75i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.256 + 0.443i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.37 + 4.25i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 + (1.48 + 2.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.67 + 2.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.05 + 7.01i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.66T + 71T^{2} \) |
| 73 | \( 1 - 3.66iT - 73T^{2} \) |
| 79 | \( 1 + (0.00584 + 0.00337i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.41 + 1.39i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.952iT - 89T^{2} \) |
| 97 | \( 1 + (-3.07 - 1.77i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.612793525102029396917264093180, −8.808459945231547802913277512839, −8.149190524581589983758717145563, −6.82820484668611407233073529878, −6.50402779157873062983223646234, −5.81157565965402295968245436685, −4.80857204037261548441810007623, −3.56829714733664325822503444198, −2.73704652022649181012157898909, −2.11303383485253431082372590778,
0.14937830672780920027123707617, 1.26675574165419551889575494781, 2.64965253881250737564230622436, 3.84642087853897937248425159119, 4.55575235053038022899411757238, 5.18008477382888059115592018714, 6.54341659652977603795197106581, 6.99066584691332286924760009417, 7.72385309402755882261147101751, 8.739001413784619125949510830356