L(s) = 1 | + (1.58 + 0.702i)3-s + (−1.53 + 2.65i)5-s + (−0.5 − 0.866i)7-s + (2.01 + 2.22i)9-s + (2.86 + 4.96i)11-s + (1.44 − 2.50i)13-s + (−4.30 + 3.13i)15-s − 4.02·17-s + 0.899·19-s + (−0.183 − 1.72i)21-s + (−1.86 + 3.22i)23-s + (−2.21 − 3.83i)25-s + (1.62 + 4.93i)27-s + (−4.43 − 7.67i)29-s + (−0.651 + 1.12i)31-s + ⋯ |
L(s) = 1 | + (0.914 + 0.405i)3-s + (−0.686 + 1.18i)5-s + (−0.188 − 0.327i)7-s + (0.670 + 0.741i)9-s + (0.864 + 1.49i)11-s + (0.401 − 0.695i)13-s + (−1.11 + 0.808i)15-s − 0.976·17-s + 0.206·19-s + (−0.0399 − 0.375i)21-s + (−0.388 + 0.672i)23-s + (−0.443 − 0.767i)25-s + (0.312 + 0.949i)27-s + (−0.823 − 1.42i)29-s + (−0.116 + 0.202i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790398747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790398747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 - 0.702i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.53 - 2.65i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 4.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.44 + 2.50i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 0.899T + 19T^{2} \) |
| 23 | \( 1 + (1.86 - 3.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.43 + 7.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.651 - 1.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + (4.99 - 8.65i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.55 - 4.41i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.40 - 9.35i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.43T + 53T^{2} \) |
| 59 | \( 1 + (-2.79 + 4.84i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.33 + 4.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.21 - 7.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 + (0.689 + 1.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.44 + 11.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.27T + 89T^{2} \) |
| 97 | \( 1 + (8.60 + 14.9i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.01695600067570304471428262037, −9.610075495943200090460981476093, −8.536382972167008133728750738178, −7.52077499011154356784091104301, −7.20311992024948704011449049076, −6.18220071016768049902837277447, −4.57343266283521995485389139637, −3.89877659060906773140887870283, −3.06764958013257817949444250087, −1.92786658026348059706303638412,
0.74654044867080083089052220714, 2.03654979697388932407349516847, 3.57231303561639118980772222792, 4.04326055594565729399058508169, 5.34315937645733339490873666199, 6.45298610668678705481797387242, 7.25298365493431205627663062725, 8.446138450494973981466946425924, 8.889546285318853332612848243889, 9.031381945753948751262992538603