| L(s) = 1 | + (−1.35 + 0.419i)2-s + (0.213 + 0.0369i)3-s + (1.64 − 1.13i)4-s + (0.614 − 1.85i)5-s + (−0.303 + 0.0395i)6-s + (−0.212 + 4.33i)7-s + (−1.74 + 2.22i)8-s + (−2.78 − 0.994i)9-s + (−0.0489 + 2.76i)10-s + (−0.695 − 0.308i)11-s + (0.393 − 0.180i)12-s + (−4.15 + 3.58i)13-s + (−1.53 − 5.94i)14-s + (0.199 − 0.373i)15-s + (1.42 − 3.73i)16-s + (1.67 − 0.894i)17-s + ⋯ |
| L(s) = 1 | + (−0.954 + 0.296i)2-s + (0.123 + 0.0213i)3-s + (0.823 − 0.566i)4-s + (0.274 − 0.831i)5-s + (−0.123 + 0.0161i)6-s + (−0.0804 + 1.63i)7-s + (−0.618 + 0.785i)8-s + (−0.926 − 0.331i)9-s + (−0.0154 + 0.875i)10-s + (−0.209 − 0.0929i)11-s + (0.113 − 0.0521i)12-s + (−1.15 + 0.994i)13-s + (−0.409 − 1.58i)14-s + (0.0515 − 0.0964i)15-s + (0.357 − 0.934i)16-s + (0.405 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.238672 + 0.489237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238672 + 0.489237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.419i)T \) |
| good | 3 | \( 1 + (-0.213 - 0.0369i)T + (2.82 + 1.01i)T^{2} \) |
| 5 | \( 1 + (-0.614 + 1.85i)T + (-4.01 - 2.97i)T^{2} \) |
| 7 | \( 1 + (0.212 - 4.33i)T + (-6.96 - 0.686i)T^{2} \) |
| 11 | \( 1 + (0.695 + 0.308i)T + (7.38 + 8.15i)T^{2} \) |
| 13 | \( 1 + (4.15 - 3.58i)T + (1.90 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 0.894i)T + (9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (5.15 - 2.91i)T + (9.76 - 16.2i)T^{2} \) |
| 23 | \( 1 + (-2.01 - 8.05i)T + (-20.2 + 10.8i)T^{2} \) |
| 29 | \( 1 + (-0.930 + 0.0228i)T + (28.9 - 1.42i)T^{2} \) |
| 31 | \( 1 + (-6.57 + 4.39i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (3.14 + 0.388i)T + (35.8 + 8.99i)T^{2} \) |
| 41 | \( 1 + (3.33 - 2.47i)T + (11.9 - 39.2i)T^{2} \) |
| 43 | \( 1 + (5.36 - 7.62i)T + (-14.4 - 40.4i)T^{2} \) |
| 47 | \( 1 + (0.350 + 3.56i)T + (-46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (1.19 + 0.0292i)T + (52.9 + 2.60i)T^{2} \) |
| 59 | \( 1 + (-1.75 + 2.03i)T + (-8.65 - 58.3i)T^{2} \) |
| 61 | \( 1 + (7.71 - 1.73i)T + (55.1 - 26.0i)T^{2} \) |
| 67 | \( 1 + (0.266 + 0.168i)T + (28.6 + 60.5i)T^{2} \) |
| 71 | \( 1 + (-10.3 - 4.88i)T + (45.0 + 54.8i)T^{2} \) |
| 73 | \( 1 + (0.000589 + 0.0120i)T + (-72.6 + 7.15i)T^{2} \) |
| 79 | \( 1 + (0.759 + 0.925i)T + (-15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (0.725 - 0.0894i)T + (80.5 - 20.1i)T^{2} \) |
| 89 | \( 1 + (2.37 - 9.47i)T + (-78.4 - 41.9i)T^{2} \) |
| 97 | \( 1 + (2.69 - 13.5i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−11.35041876166945287567834706474, −9.813407833940739470359811190834, −9.354555153866332278633618882140, −8.639361141967146314595320317318, −8.002582943083489590358723679756, −6.60596710835909591399202894452, −5.71556796313909676757939365325, −5.02593500270993570051481012562, −2.88020121530893028635194815749, −1.83428182484140455740047201615,
0.40059135722321828856843075625, 2.43903567179131770183020767408, 3.23652412981193252638009260675, 4.76523087910311664034587718346, 6.44656658673925512660866670093, 7.04384359404760197689760136550, 7.950908471418135457801538782839, 8.705060771400853044098675460195, 10.18224707238492388871948322345, 10.40885740578503755611573846293
