L(s) = 1 | − i·3-s − 3i·5-s − 3·7-s + 2·9-s − i·13-s − 3·15-s − 7·17-s − 4i·19-s + 3i·21-s − 4·23-s − 4·25-s − 5i·27-s + 4i·29-s + 8·31-s + 9i·35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.34i·5-s − 1.13·7-s + 0.666·9-s − 0.277i·13-s − 0.774·15-s − 1.69·17-s − 0.917i·19-s + 0.654i·21-s − 0.834·23-s − 0.800·25-s − 0.962i·27-s + 0.742i·29-s + 1.43·31-s + 1.52i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.389250 - 0.939732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.389250 - 0.939732i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.88697112495268797091923845999, −9.752886466224685561337253962274, −9.067145886263812370717231823172, −8.212080707638445255604324338111, −7.00977517809252333308656063861, −6.29294867830547392975854592743, −4.96686704061183388838479542014, −4.01260748526645672830388922689, −2.29119443694623589441939074338, −0.64360083432287352970178500465,
2.41091872202386334518782012665, 3.56791279260911188479866218575, 4.46974671565504062997968650541, 6.25777464968427212796214593142, 6.61711645796808404555812741294, 7.73026060178007962197194849978, 9.089775066662220953126225337654, 10.01564076271265384887889462698, 10.38826353783580439289590554285, 11.36169354213970672628326571399