L(s) = 1 | + (0.342 + 0.939i)2-s + (−1.85 − 0.326i)3-s + (−0.766 + 0.642i)4-s + (−0.326 − 1.85i)6-s + (−0.866 − 0.500i)8-s + (2.37 + 0.866i)9-s + (−0.173 + 0.300i)11-s + (1.62 − 0.939i)12-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s + 2.53i·18-s + (−0.766 + 0.642i)19-s + (−0.342 − 0.0603i)22-s + (1.43 + 1.20i)24-s + (−2.49 − 1.43i)27-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)2-s + (−1.85 − 0.326i)3-s + (−0.766 + 0.642i)4-s + (−0.326 − 1.85i)6-s + (−0.866 − 0.500i)8-s + (2.37 + 0.866i)9-s + (−0.173 + 0.300i)11-s + (1.62 − 0.939i)12-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s + 2.53i·18-s + (−0.766 + 0.642i)19-s + (−0.342 − 0.0603i)22-s + (1.43 + 1.20i)24-s + (−2.49 − 1.43i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5935739531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5935739531\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
good | 3 | \( 1 + (1.85 + 0.326i)T + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \) |
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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
−8.582040678520319802131286734954, −7.57426467832171181672200299551, −7.09896268396477670694328100812, −6.49411107694637236246842180558, −5.75277749803782949682201671207, −5.25898141692672780032503732565, −4.53212397580944356946099465971, −3.81403383829402909260491181492, −2.18439565928928402818712760306, −0.59516189190521390542563173250,
0.78598950242357239580501925649, 1.91323602548740400020412953216, 3.24118834792484319112290331844, 4.37220743375598972766409215075, 4.58397158790654365499849287908, 5.58031559897304795894436128648, 6.12574402520474266006163036162, 6.67579645722138511426426769328, 7.87113216080578245728752722616, 8.901113787554589020123753822949