L(s) = 1 | + (−2 − 1.73i)7-s + 5·13-s + (−4 − 6.92i)19-s + (2.5 − 4.33i)25-s + (3.5 − 6.06i)31-s + (−5.5 − 9.52i)37-s + 5·43-s + (1.00 + 6.92i)49-s + (6.5 + 11.2i)61-s + (−2.5 + 4.33i)67-s + (5 − 8.66i)73-s + (−8.5 − 14.7i)79-s + (−10 − 8.66i)91-s + 5·97-s + (6.5 + 11.2i)103-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)7-s + 1.38·13-s + (−0.917 − 1.58i)19-s + (0.5 − 0.866i)25-s + (0.628 − 1.08i)31-s + (−0.904 − 1.56i)37-s + 0.762·43-s + (0.142 + 0.989i)49-s + (0.832 + 1.44i)61-s + (−0.305 + 0.529i)67-s + (0.585 − 1.01i)73-s + (−0.956 − 1.65i)79-s + (−1.04 − 0.907i)91-s + 0.507·97-s + (0.640 + 1.10i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01043 - 0.768695i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01043 - 0.768695i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−10.36085352411742237833569783296, −9.183872721975917997357219882368, −8.639266726384180833585740260978, −7.47428265685463010285386319793, −6.61078330450288838466668901466, −5.93539379745497448383189653466, −4.52859105018188863138849443064, −3.71449201499983786824730959351, −2.48793272427293153632119263446, −0.68110339004898393384495676703,
1.55263706632232339358065974440, 3.06720014594391318978330541493, 3.91089356611170728681137737208, 5.29954957061760091761372181978, 6.17631029587674796795130793101, 6.81585722559007291562249764007, 8.242672039494663270255201750844, 8.666957185865336672839115854772, 9.706371124018261947577911616879, 10.47051467324555825501408218195