L(s) = 1 | + 3.37i·3-s − i·7-s − 8.37·9-s + 0.627·11-s + 1.37i·13-s + 5.37i·17-s − 6.74·19-s + 3.37·21-s − 6.74i·23-s − 18.1i·27-s − 1.37·29-s − 8·31-s + 2.11i·33-s − 2i·37-s − 4.62·39-s + ⋯ |
L(s) = 1 | + 1.94i·3-s − 0.377i·7-s − 2.79·9-s + 0.189·11-s + 0.380i·13-s + 1.30i·17-s − 1.54·19-s + 0.735·21-s − 1.40i·23-s − 3.48i·27-s − 0.254·29-s − 1.43·31-s + 0.368i·33-s − 0.328i·37-s − 0.741·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4393066425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4393066425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3.37iT - 3T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 1.37iT - 13T^{2} \) |
| 17 | \( 1 - 5.37iT - 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 + 2.74iT - 43T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 - 0.744iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 18.8iT - 97T^{2} \) |
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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
−10.24075995456023531810391309424, −9.314869343567223783006304152522, −8.737862012945496801387947842569, −8.064371436921846099171193998787, −6.58114040421040890928974585549, −5.86976570282921137780113610587, −4.81562663298553157029220312574, −4.13403506249881223823261885110, −3.57897200567973441117360518701, −2.24573832105176853854466750475,
0.16691496737015332320128859463, 1.58908731437478668418211975694, 2.42422320143556882085923600910, 3.47034624048576921377016098199, 5.18028005374838866352458891943, 5.84558205267378469929778189905, 6.77517282403064241343994874454, 7.26635584776723341949151735547, 8.114455864888384226593391780250, 8.752319938735178038949507641643