L(s) = 1 | + (−0.933 + 1.61i)2-s + (−1.67 − 2.90i)3-s + (−0.741 − 1.28i)4-s + (0.433 − 0.750i)5-s + 6.24·6-s − 0.965·8-s + (−4.10 + 7.11i)9-s + (0.808 + 1.39i)10-s + (1.93 + 3.34i)11-s + (−2.48 + 4.30i)12-s − 13-s − 2.90·15-s + (2.38 − 4.12i)16-s + (−1.67 − 2.90i)17-s + (−7.66 − 13.2i)18-s + (−2.69 + 4.66i)19-s + ⋯ |
L(s) = 1 | + (−0.659 + 1.14i)2-s + (−0.966 − 1.67i)3-s + (−0.370 − 0.642i)4-s + (0.193 − 0.335i)5-s + 2.55·6-s − 0.341·8-s + (−1.36 + 2.37i)9-s + (0.255 + 0.442i)10-s + (0.582 + 1.00i)11-s + (−0.716 + 1.24i)12-s − 0.277·13-s − 0.748·15-s + (0.595 − 1.03i)16-s + (−0.406 − 0.703i)17-s + (−1.80 − 3.12i)18-s + (−0.617 + 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.558379 + 0.276803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.558379 + 0.276803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.933 - 1.61i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.67 + 2.90i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.433 + 0.750i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 - 3.34i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.67 + 2.90i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.62 + 4.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + (-3.78 - 6.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.41 + 4.18i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + (1.82 - 3.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.107 - 0.186i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.51 + 7.82i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 - 6.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 + (-7.77 - 13.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.71 - 8.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 + (0.209 - 0.362i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.11T + 97T^{2} \) |
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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
−10.77238161355547631156170490771, −9.563386097196786302463286413794, −8.571889387278216155333144564731, −7.84735429004239104682497785957, −6.91027664995628582688884658132, −6.69503685015007181611022342004, −5.68603173571458807222172193263, −4.81116395692576187879968197717, −2.39634611083228809038662027370, −1.00785018681790447091423011225,
0.61706031465123877403041645720, 2.73043287485785794978781421092, 3.70627955530079397523053057805, 4.64691595290980830789121951508, 5.90952704591836929010279592334, 6.46527398635974966112049301849, 8.496625317591681759246469725980, 9.169945780335559196990219111927, 9.808269438410263241105193237312, 10.57854538336156087723174121776