L(s) = 1 | + (−0.704 + 0.256i)2-s + (0.294 − 1.67i)3-s + (−0.336 + 0.281i)4-s + (0.220 + 1.25i)6-s + (−0.997 − 1.72i)7-s + (0.538 − 0.933i)8-s + (−1.76 − 0.641i)9-s + (0.5 − 0.866i)11-s + (0.372 + 0.644i)12-s + (−0.173 − 0.984i)13-s + (1.14 + 0.960i)14-s + (−0.0640 + 0.363i)16-s + 1.40·18-s + (0.882 − 0.469i)19-s + (−3.17 + 1.15i)21-s + (−0.130 + 0.737i)22-s + ⋯ |
L(s) = 1 | + (−0.704 + 0.256i)2-s + (0.294 − 1.67i)3-s + (−0.336 + 0.281i)4-s + (0.220 + 1.25i)6-s + (−0.997 − 1.72i)7-s + (0.538 − 0.933i)8-s + (−1.76 − 0.641i)9-s + (0.5 − 0.866i)11-s + (0.372 + 0.644i)12-s + (−0.173 − 0.984i)13-s + (1.14 + 0.960i)14-s + (−0.0640 + 0.363i)16-s + 1.40·18-s + (0.882 − 0.469i)19-s + (−3.17 + 1.15i)21-s + (−0.130 + 0.737i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6925744514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6925744514\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-0.882 + 0.469i)T \) |
good | 2 | \( 1 + (0.704 - 0.256i)T + (0.766 - 0.642i)T^{2} \) |
| 3 | \( 1 + (-0.294 + 1.67i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.997 + 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-1.47 + 1.23i)T + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.194 - 1.10i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.0534 + 0.0448i)T + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.241 + 0.419i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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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
−8.460908566100001911613754732679, −7.75881881663070393191640549874, −7.11852452913230438063685442418, −6.88156721066162718496196831450, −6.01857622370776221329259858027, −4.66780534965741679341029785540, −3.36460270635726673860180272119, −3.06548581307168029416230445041, −1.08227980969358040978710317496, −0.69120150386144275968211190354,
1.92847961337346458341436017613, 2.89392507396070023866349931144, 3.78561910821738734533704110148, 4.79587431981827564506954454038, 5.29826775245983227944692453889, 6.09956775014749025130984315609, 7.24139687151277443545736317789, 8.532032462961314382934868770412, 8.970122167638909142518201231783, 9.492911259230214549583903062635