L(s) = 1 | − 1.73i·5-s + 1.73i·11-s − 5.19i·17-s − 7·19-s + 8.66i·23-s + 2.00·25-s + 6·29-s − 5·31-s − 5·37-s + 6.92i·41-s − 3.46i·43-s + 3·47-s + 9·53-s + 2.99·55-s + 9·59-s + ⋯ |
L(s) = 1 | − 0.774i·5-s + 0.522i·11-s − 1.26i·17-s − 1.60·19-s + 1.80i·23-s + 0.400·25-s + 1.11·29-s − 0.898·31-s − 0.821·37-s + 1.08i·41-s − 0.528i·43-s + 0.437·47-s + 1.23·53-s + 0.404·55-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.633389327\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633389327\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.026475907570667547216724708315, −7.19571912842469547367719109107, −6.75848535327920512346273789894, −5.70676722632084493973262372871, −5.13120895994906970240047178107, −4.47761594457905485660716018121, −3.72429222487841975771294925665, −2.69399820449790123844939848719, −1.81435211031509406449427572549, −0.797655410376273822697492592943,
0.51862816290019674256169529816, 1.95241034889330119870062566236, 2.60375618833385925078606604664, 3.57254945916163737161655319805, 4.19005759470164429182672936952, 5.05655078076346541028736057769, 6.04315310811076425445779523108, 6.50369004015251591324545535905, 7.00866407414404270157804879262, 8.028428612603451586522287380622