L(s) = 1 | + (−1.41 + 1.41i)4-s + (1.14 − 2.77i)9-s + (1.25 − 1.87i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (1.05 + 3.98i)17-s + (1.81 + 1.20i)23-s + (4.61 + 1.91i)25-s + (4.54 + 6.80i)31-s + (2.29 + 5.54i)36-s + (0.646 − 0.128i)41-s + (−2.50 + 6.05i)43-s + (0.881 + 4.43i)44-s + (9.65 + 9.65i)47-s + (6.46 − 2.67i)49-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)4-s + (0.382 − 0.923i)9-s + (0.378 − 0.566i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (0.255 + 0.966i)17-s + (0.377 + 0.252i)23-s + (0.923 + 0.382i)25-s + (0.816 + 1.22i)31-s + (0.382 + 0.923i)36-s + (0.100 − 0.0200i)41-s + (−0.382 + 0.923i)43-s + (0.132 + 0.668i)44-s + (1.40 + 1.40i)47-s + (0.923 − 0.382i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33445 + 0.340206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33445 + 0.340206i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-1.05 - 3.98i)T \) |
| 43 | \( 1 + (2.50 - 6.05i)T \) |
good | 2 | \( 1 + (1.41 - 1.41i)T^{2} \) |
| 3 | \( 1 + (-1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 1.87i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \) |
| 19 | \( 1 + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.81 - 1.20i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-4.54 - 6.80i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-0.646 + 0.128i)T + (37.8 - 15.6i)T^{2} \) |
| 47 | \( 1 + (-9.65 - 9.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.55 + 13.4i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (2.19 + 0.907i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + 0.318iT - 67T^{2} \) |
| 71 | \( 1 + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-9.76 + 14.6i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-6.05 + 2.50i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + 89iT^{2} \) |
| 97 | \( 1 + (3.29 - 16.5i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−10.40868846878331494856580391038, −9.338583343719414413921160687533, −8.841163097743187497989396455366, −8.038978921078566567903978610865, −6.92904537646876789841808412098, −6.14228900557983035870258799583, −4.85229660013148388983106097127, −3.85894418444215360449061594518, −3.16617432962985955057727007375, −1.15092702876879522838142360742,
0.975998848378082664600197881796, 2.46791567754583328729401503250, 4.07648030418838397876149265938, 4.88225587279996787045179685172, 5.68923439234557666924014898304, 6.81568463309991933336935267367, 7.76779833076862585210715050449, 8.735684106720091775943961910662, 9.504238254398560152765831045905, 10.35050015020776566861677161989