L(s) = 1 | + (−0.555 + 0.962i)5-s + (−0.5 − 0.866i)7-s + (−0.944 − 1.63i)11-s + (0.5 − 0.866i)13-s + 5.87·17-s + 7.09·19-s + (1.99 − 3.45i)23-s + (1.88 + 3.26i)25-s + (0.493 + 0.855i)29-s + (0.333 − 0.576i)31-s + 1.11·35-s − 1.33·37-s + (−0.944 + 1.63i)41-s + (5.43 + 9.40i)43-s + (−5.54 − 9.61i)47-s + ⋯ |
L(s) = 1 | + (−0.248 + 0.430i)5-s + (−0.188 − 0.327i)7-s + (−0.284 − 0.493i)11-s + (0.138 − 0.240i)13-s + 1.42·17-s + 1.62·19-s + (0.415 − 0.720i)23-s + (0.376 + 0.652i)25-s + (0.0916 + 0.158i)29-s + (0.0598 − 0.103i)31-s + 0.187·35-s − 0.219·37-s + (−0.147 + 0.255i)41-s + (0.828 + 1.43i)43-s + (−0.809 − 1.40i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49614 - 0.109188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49614 - 0.109188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.555 - 0.962i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.944 + 1.63i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 + (-1.99 + 3.45i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.493 - 0.855i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.333 + 0.576i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 + (0.944 - 1.63i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.43 - 9.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.54 + 9.61i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + (-2.38 + 4.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.88 + 3.26i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.04 - 3.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + (3.21 + 5.56i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + (0.382 + 0.662i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.28060994655989224768052021554, −9.643449559664113188462117705235, −8.529065751401570212675231171134, −7.65128364008286494720398604368, −6.99445962458058099268646734492, −5.83526474972632247702276043499, −5.02162545401017479031138444263, −3.57619716120214753988244862522, −2.94284488232426332824125288468, −1.01942786560611794143905654828,
1.16255006564115699392039085438, 2.78223821277526226933979136015, 3.86035194981524524969622361774, 5.10166165030897111845489729637, 5.71962756268206802314002087600, 7.05905396955844962605767671946, 7.73435118983721620045315696107, 8.685448195444007063152381128524, 9.569406022924669278628481486173, 10.17962826535728632116097585601