L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.420 − 2.92i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (−1.22 + 2.68i)6-s + (2.73 − 3.15i)7-s + (0.142 − 0.989i)8-s + (−5.51 + 1.61i)9-s + (0.654 + 0.755i)10-s + (0.283 − 0.181i)11-s + (2.48 − 1.59i)12-s + (−2.21 − 2.55i)13-s + (−4.00 + 1.17i)14-s + (−0.420 + 2.92i)15-s + (−0.654 + 0.755i)16-s + (−1.82 + 4.00i)17-s + ⋯ |
L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.242 − 1.68i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.501 + 1.09i)6-s + (1.03 − 1.19i)7-s + (0.0503 − 0.349i)8-s + (−1.83 + 0.539i)9-s + (0.207 + 0.238i)10-s + (0.0853 − 0.0548i)11-s + (0.718 − 0.461i)12-s + (−0.614 − 0.709i)13-s + (−1.07 + 0.314i)14-s + (−0.108 + 0.755i)15-s + (−0.163 + 0.188i)16-s + (−0.443 + 0.970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0496074 - 0.753970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0496074 - 0.753970i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (2.79 - 3.89i)T \) |
good | 3 | \( 1 + (0.420 + 2.92i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 3.15i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.283 + 0.181i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.21 + 2.55i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.82 - 4.00i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.00 - 4.39i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.04 + 6.67i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.759 + 5.28i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.80 + 2.29i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.0346 - 0.0101i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.80 + 12.5i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 + (0.0676 - 0.0780i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (5.00 + 5.77i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.30 - 9.06i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-7.18 - 4.62i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-12.6 - 8.12i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.21 + 4.85i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.98 - 10.3i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (0.0754 - 0.0221i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.30 + 9.08i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.45 + 1.01i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.76179846710063799431243742020, −11.03537823210879181735011804110, −9.990524501299087952987020222857, −8.181404783531351038104572510130, −7.889876570728448873951827528986, −7.11660990802739391955752821860, −5.79750748909259443031868996903, −4.01803361250208661931879739361, −2.05789050294137810859155999163, −0.801115124181985169879200970705,
2.73824740973624178002003074009, 4.68689447235018757151225930208, 5.03254553173133636974224381860, 6.54617245650447114792704892939, 8.046727879007272242815447194990, 9.043736939668890143920653592079, 9.494954438142335907282442200273, 10.76226219376225094219723066526, 11.40352474728181780676613947304, 12.10858583123265403091910780545