L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + (1 + 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s − 0.999·12-s + (3.5 + 0.866i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + 0.999·18-s + (−1 − 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s − 0.288·12-s + (0.970 + 0.240i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + 0.235·18-s + (−0.229 − 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8432620672\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8432620672\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.296293690340900925765819938554, −8.643717069149459106011447601934, −7.921729590370679079173151828655, −7.19750935368901669164601888926, −6.61878459451499097519724862605, −5.48689662705316968474346515922, −4.96379307788970275424519328362, −3.76051256825885755036106234753, −2.36785093839935018985393230343, −1.60552103111014607058052376755,
0.31905442401513903992629206416, 1.73483325226765150085499433309, 2.98608368215299611266035932527, 3.71382677890955821401430475414, 4.49192324638086014736300277949, 5.62554748738814150318699852792, 6.37676094692263901996284236351, 7.74853050479598642576672807887, 8.274442127281453992446882320501, 8.715927878463112715413151348196