L(s) = 1 | + (−1.43 + 1.20i)11-s + (0.173 − 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (−0.266 + 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (1.17 + 0.984i)59-s + (0.326 − 1.85i)67-s + (−0.173 − 0.300i)73-s + (−0.173 − 0.984i)83-s + (−0.5 − 0.866i)89-s + (−1.43 + 1.20i)97-s + 0.347·107-s + (0.766 + 0.642i)113-s + ⋯ |
L(s) = 1 | + (−1.43 + 1.20i)11-s + (0.173 − 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (−0.266 + 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (1.17 + 0.984i)59-s + (0.326 − 1.85i)67-s + (−0.173 − 0.300i)73-s + (−0.173 − 0.984i)83-s + (−0.5 − 0.866i)89-s + (−1.43 + 1.20i)97-s + 0.347·107-s + (0.766 + 0.642i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9906048754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9906048754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.385194169726431946057527077303, −8.302894055649512105576760354994, −7.63382943529902681853348281556, −7.17943379888656645000945311835, −6.04201288044513282668208886836, −5.26259457165386812163072714609, −4.63668917642408023719454921272, −3.50150686272382424135787698450, −2.59891757200664342494343042608, −1.51146481085355359407134417322,
0.65642082288869425926984251620, 2.35623851475733170797214501753, 3.06809177748042540077109202365, 4.07969017726788857856444149683, 5.25369891368655717889394318999, 5.56917451837576997165904534345, 6.69487609997346532984228161211, 7.38345932621670333068301091112, 8.348889001260723865903804531257, 8.646761586536275837947463794328