| L(s) = 1 | + (0.938 + 1.05i)2-s + (−0.239 + 1.98i)4-s + (1.55 + 1.84i)5-s + (−1.49 + 0.544i)7-s + (−2.32 + 1.60i)8-s + (−0.500 + 3.37i)10-s + (−0.102 + 0.122i)11-s + (−3.39 + 0.597i)13-s + (−1.98 − 1.07i)14-s + (−3.88 − 0.950i)16-s + (1.43 + 2.48i)17-s + (3.75 + 2.16i)19-s + (−4.04 + 2.63i)20-s + (−0.225 + 0.00614i)22-s + (−2.14 − 0.781i)23-s + ⋯ |
| L(s) = 1 | + (0.663 + 0.748i)2-s + (−0.119 + 0.992i)4-s + (0.693 + 0.826i)5-s + (−0.565 + 0.205i)7-s + (−0.822 + 0.569i)8-s + (−0.158 + 1.06i)10-s + (−0.0308 + 0.0367i)11-s + (−0.940 + 0.165i)13-s + (−0.529 − 0.286i)14-s + (−0.971 − 0.237i)16-s + (0.348 + 0.603i)17-s + (0.861 + 0.497i)19-s + (−0.903 + 0.589i)20-s + (−0.0480 + 0.00131i)22-s + (−0.447 − 0.162i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.586309 + 1.84755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586309 + 1.84755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.938 - 1.05i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.55 - 1.84i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.49 - 0.544i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.102 - 0.122i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (3.39 - 0.597i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.43 - 2.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 2.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.14 + 0.781i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.60 - 1.51i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (6.69 + 2.43i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.14 + 2.97i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.74 - 9.90i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.59 + 9.05i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (3.66 - 1.33i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 9.80iT - 53T^{2} \) |
| 59 | \( 1 + (-1.72 - 2.05i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.492 - 1.35i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.225 - 0.0397i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.82 + 4.88i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.14 + 12.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.18 - 12.3i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 1.95i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.73 + 6.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.17 - 4.34i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.87165515091330662587718794631, −9.922302857140260812690959947051, −9.235390673456845214257859675480, −7.983294267632272157347706167191, −7.21864737549667311173645037818, −6.28248403230848509295080682219, −5.74873851696142730620951299161, −4.54846503437229047625512903452, −3.29908779292171173612484769416, −2.40045002347543398192224919865,
0.849395236075064704918879592608, 2.32969877473694584381105468610, 3.41868352948175828576933375221, 4.77296328947189078571112164509, 5.32977398194496148896779478544, 6.35537038156399294109784934204, 7.44002606959181105784676900293, 8.848768571129569247104355667342, 9.704054866845811577498175480725, 9.985030417588907454125412356263
