L(s) = 1 | + (−0.923 + 0.382i)3-s + (0.155 − 0.375i)5-s + (0.709 + 0.709i)7-s + (0.707 − 0.707i)9-s + (2.79 + 1.15i)11-s + (−2.58 − 6.24i)13-s + 0.406i·15-s − 1.05i·17-s + (1.48 + 3.59i)19-s + (−0.926 − 0.383i)21-s + (0.922 − 0.922i)23-s + (3.41 + 3.41i)25-s + (−0.382 + 0.923i)27-s + (7.64 − 3.16i)29-s − 1.88·31-s + ⋯ |
L(s) = 1 | + (−0.533 + 0.220i)3-s + (0.0696 − 0.168i)5-s + (0.268 + 0.268i)7-s + (0.235 − 0.235i)9-s + (0.841 + 0.348i)11-s + (−0.717 − 1.73i)13-s + 0.105i·15-s − 0.256i·17-s + (0.341 + 0.823i)19-s + (−0.202 − 0.0837i)21-s + (0.192 − 0.192i)23-s + (0.683 + 0.683i)25-s + (−0.0736 + 0.177i)27-s + (1.41 − 0.587i)29-s − 0.338·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37184 - 0.118917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37184 - 0.118917i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
good | 5 | \( 1 + (-0.155 + 0.375i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.709 - 0.709i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.79 - 1.15i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (2.58 + 6.24i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 1.05iT - 17T^{2} \) |
| 19 | \( 1 + (-1.48 - 3.59i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.922 + 0.922i)T - 23iT^{2} \) |
| 29 | \( 1 + (-7.64 + 3.16i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 + (1.24 - 3.01i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.11 + 5.11i)T - 41iT^{2} \) |
| 43 | \( 1 + (-10.9 - 4.53i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 7.47iT - 47T^{2} \) |
| 53 | \( 1 + (-7.58 - 3.14i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.13 + 9.98i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (1.35 - 0.562i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (10.8 - 4.50i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-9.35 - 9.35i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.367 + 0.367i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.87iT - 79T^{2} \) |
| 83 | \( 1 + (1.62 + 3.91i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-8.33 - 8.33i)T + 89iT^{2} \) |
| 97 | \( 1 + 7.97T + 97T^{2} \) |
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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
−10.26784862178551804554959301282, −9.616039580441616581289900902282, −8.615839330262723936955709185093, −7.70201741320640276752208232060, −6.78782631273619560392562339065, −5.66236647602624848316176677546, −5.07999110699398112211561131412, −3.93687777797677091606417906982, −2.65707209754993718739635616666, −0.968089744250196367106200226359,
1.16589151008672316964026115705, 2.57081986864185707591790168646, 4.14319922698203566361825292054, 4.82091890996554140979451164513, 6.12295902600446759455920046828, 6.82606259060133317512885396998, 7.50875608825585301558146197160, 8.855575061657207717410250099392, 9.363958382474051992084805619246, 10.54236021591503363467998419623