L(s) = 1 | + (−1.17 + 2.02i)2-s + (−0.573 − 0.992i)3-s + (−1.74 − 3.02i)4-s + (−0.671 + 1.16i)5-s + 2.68·6-s + 3.48·8-s + (0.842 − 1.45i)9-s + (−1.57 − 2.72i)10-s + (−0.573 − 0.992i)11-s + (−2 + 3.46i)12-s − 13-s + 1.53·15-s + (−0.598 + 1.03i)16-s + (2.91 + 5.05i)17-s + (1.97 + 3.42i)18-s + (−1.67 + 2.89i)19-s + ⋯ |
L(s) = 1 | + (−0.828 + 1.43i)2-s + (−0.330 − 0.573i)3-s + (−0.872 − 1.51i)4-s + (−0.300 + 0.520i)5-s + 1.09·6-s + 1.23·8-s + (0.280 − 0.486i)9-s + (−0.497 − 0.861i)10-s + (−0.172 − 0.299i)11-s + (−0.577 + 0.999i)12-s − 0.277·13-s + 0.397·15-s + (−0.149 + 0.259i)16-s + (0.707 + 1.22i)17-s + (0.465 + 0.806i)18-s + (−0.383 + 0.664i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.244836 + 0.584218i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.244836 + 0.584218i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.17 - 2.02i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.573 + 0.992i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.671 - 1.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.573 + 0.992i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.67 - 2.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + (-0.817 - 1.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.25 - 7.37i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.292T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + (5.30 - 9.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.391 - 0.677i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.32 - 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 5.28i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 73 | \( 1 + (7.65 + 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.441 - 0.764i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−10.47173094350785399255824783372, −10.02078878661879746114151564182, −8.770754503813991018880245142019, −8.139074716891102147140924126146, −7.32767970424370337777756659244, −6.50371526121381927110675508856, −6.09359503305258686717106588979, −4.84146415692001326045193405397, −3.29570918993894927309806988445, −1.17963907157478417566298123936,
0.56694498645074791712338353965, 2.13434543047120394955402936252, 3.30609991253543062908775183409, 4.53804492631684499752186101669, 5.18039264175491817948144228020, 6.95866467109799522294907380591, 8.046506964060252610511468017143, 8.788872299741759033649580021559, 9.793496979047413818607577280001, 10.12972442396120312253546962044