L(s) = 1 | + (1.48 + 1.48i)2-s + (1.53 − 0.809i)3-s + 2.42i·4-s + (−0.854 − 0.854i)5-s + (3.48 + 1.07i)6-s + (0.707 + 0.707i)7-s + (−0.634 + 0.634i)8-s + (1.69 − 2.47i)9-s − 2.54i·10-s + (−0.183 + 0.183i)11-s + (1.96 + 3.71i)12-s + (−2.40 + 2.68i)13-s + 2.10i·14-s + (−1.99 − 0.616i)15-s + 2.96·16-s − 7.60·17-s + ⋯ |
L(s) = 1 | + (1.05 + 1.05i)2-s + (0.884 − 0.467i)3-s + 1.21i·4-s + (−0.382 − 0.382i)5-s + (1.42 + 0.438i)6-s + (0.267 + 0.267i)7-s + (−0.224 + 0.224i)8-s + (0.563 − 0.826i)9-s − 0.803i·10-s + (−0.0554 + 0.0554i)11-s + (0.566 + 1.07i)12-s + (−0.666 + 0.745i)13-s + 0.562i·14-s + (−0.516 − 0.159i)15-s + 0.741·16-s − 1.84·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.35532 + 1.04569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.35532 + 1.04569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.53 + 0.809i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (2.40 - 2.68i)T \) |
good | 2 | \( 1 + (-1.48 - 1.48i)T + 2iT^{2} \) |
| 5 | \( 1 + (0.854 + 0.854i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.183 - 0.183i)T - 11iT^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 + (4.33 - 4.33i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.600T + 23T^{2} \) |
| 29 | \( 1 - 2.36iT - 29T^{2} \) |
| 31 | \( 1 + (-6.76 + 6.76i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.20 + 2.20i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.79 - 2.79i)T + 41iT^{2} \) |
| 43 | \( 1 + 2.48iT - 43T^{2} \) |
| 47 | \( 1 + (5.78 - 5.78i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.27iT - 53T^{2} \) |
| 59 | \( 1 + (-7.52 + 7.52i)T - 59iT^{2} \) |
| 61 | \( 1 + 0.457T + 61T^{2} \) |
| 67 | \( 1 + (5.51 - 5.51i)T - 67iT^{2} \) |
| 71 | \( 1 + (-9.60 - 9.60i)T + 71iT^{2} \) |
| 73 | \( 1 + (-4.31 - 4.31i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.20T + 79T^{2} \) |
| 83 | \( 1 + (0.439 + 0.439i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.82 - 3.82i)T - 89iT^{2} \) |
| 97 | \( 1 + (-5.33 + 5.33i)T - 97iT^{2} \) |
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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
−12.55122860371349215042545175498, −11.50322491808310823701243298388, −9.906609791997304580012333306660, −8.663997790578583332466449008762, −8.034293161148653910578435610449, −6.94043493827132621803297873178, −6.23780092569875326577837697416, −4.65886748098819869500023189393, −4.03580859664910497041547154181, −2.25787815656650149222539342521,
2.23462794010422435791085670837, 3.14020630769958923458377523977, 4.31907326061106914449290055465, 4.96357741639806334382349829679, 6.82336178734995439195082153961, 8.014137819731075019490949306725, 9.038817874894698948966557681195, 10.37779418125062666643768577287, 10.82287854595217627582015453241, 11.73667125447120349598686541774