L(s) = 1 | + (−1.22 − 1.22i)3-s + (4.54 + 4.54i)5-s + 1.15·7-s + 2.99i·9-s + (−6.42 + 6.42i)11-s + (14.8 − 14.8i)13-s − 11.1i·15-s + 15.0·17-s + (9.44 + 9.44i)19-s + (−1.41 − 1.41i)21-s − 31.6·23-s + 16.3i·25-s + (3.67 − 3.67i)27-s + (−5.51 + 5.51i)29-s + 20.3i·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.909 + 0.909i)5-s + 0.164·7-s + 0.333i·9-s + (−0.584 + 0.584i)11-s + (1.14 − 1.14i)13-s − 0.742i·15-s + 0.887·17-s + (0.497 + 0.497i)19-s + (−0.0671 − 0.0671i)21-s − 1.37·23-s + 0.653i·25-s + (0.136 − 0.136i)27-s + (−0.190 + 0.190i)29-s + 0.656i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.951095590\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.951095590\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-4.54 - 4.54i)T + 25iT^{2} \) |
| 7 | \( 1 - 1.15T + 49T^{2} \) |
| 11 | \( 1 + (6.42 - 6.42i)T - 121iT^{2} \) |
| 13 | \( 1 + (-14.8 + 14.8i)T - 169iT^{2} \) |
| 17 | \( 1 - 15.0T + 289T^{2} \) |
| 19 | \( 1 + (-9.44 - 9.44i)T + 361iT^{2} \) |
| 23 | \( 1 + 31.6T + 529T^{2} \) |
| 29 | \( 1 + (5.51 - 5.51i)T - 841iT^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 + (-50.3 - 50.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 52.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-42.2 + 42.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 27.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.1 + 20.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-69.0 + 69.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-0.992 + 0.992i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-77.0 - 77.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 44.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 3.56iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-90.2 - 90.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 68.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 161.T + 9.40e3T^{2} \) |
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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
−10.17137454654096047593017527334, −9.770536195505864974892366866447, −8.184668400346774461894785705285, −7.71238329028749018171023063093, −6.49967280837067835901788648885, −5.92466365395531139931432841848, −5.11727615697647253717032446511, −3.52683908301921966409575883469, −2.46696692261242036466893927579, −1.22791465955566149345195312812,
0.820115918800553368676051403867, 2.08197207200521400514770304638, 3.70729152785322005207477281873, 4.66990185852074925139328752320, 5.78915804196590835716909071380, 6.00639202449894604217323702481, 7.52222278108399610774764310091, 8.528507184292263931726168567406, 9.295693596946004692088644093595, 9.875884636621781855386320206023