L(s) = 1 | − 2.19·3-s − 0.364·7-s + 1.83·9-s − 1.80·11-s − 2.03·13-s + 7.66·17-s − 4·19-s + 0.801·21-s + 4·23-s − 5·25-s + 2.56·27-s − 8.06·29-s + 3.27·31-s + 3.96·33-s − 6.79·37-s + 4.46·39-s + 11.7·41-s + 0.105·43-s − 0.768·47-s − 6.86·49-s − 16.8·51-s − 6.46·53-s + 8.79·57-s + 7.23·59-s + 2·61-s − 0.668·63-s − 13.5·67-s + ⋯ |
L(s) = 1 | − 1.26·3-s − 0.137·7-s + 0.611·9-s − 0.543·11-s − 0.563·13-s + 1.85·17-s − 0.917·19-s + 0.174·21-s + 0.834·23-s − 25-s + 0.493·27-s − 1.49·29-s + 0.587·31-s + 0.689·33-s − 1.11·37-s + 0.715·39-s + 1.83·41-s + 0.0160·43-s − 0.112·47-s − 0.981·49-s − 2.36·51-s − 0.887·53-s + 1.16·57-s + 0.941·59-s + 0.256·61-s − 0.0842·63-s − 1.65·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7288552427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7288552427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 0.364T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.105T + 43T^{2} \) |
| 47 | \( 1 + 0.768T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.331T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.364T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.69831708414077183758509847466, −7.11573368547362943992496685343, −6.27996132661791284930389219127, −5.66187821473602796440339282645, −5.26433961887445022132788178078, −4.48409079970118505397411819494, −3.56438597147690851387005176642, −2.68880451136526008451990569556, −1.57025990380170413415093403700, −0.45715724949283147624762282830,
0.45715724949283147624762282830, 1.57025990380170413415093403700, 2.68880451136526008451990569556, 3.56438597147690851387005176642, 4.48409079970118505397411819494, 5.26433961887445022132788178078, 5.66187821473602796440339282645, 6.27996132661791284930389219127, 7.11573368547362943992496685343, 7.69831708414077183758509847466