L(s) = 1 | + 4·5-s + 2·11-s + 6·13-s − 4·17-s − 4·19-s + 2·23-s + 11·25-s + 2·29-s + 2·37-s + 4·43-s − 12·47-s + 6·53-s + 8·55-s + 8·59-s − 6·61-s + 24·65-s + 8·67-s + 14·71-s + 2·73-s − 12·79-s + 4·83-s − 16·85-s − 16·95-s + 2·97-s + 16·101-s − 16·103-s + 18·107-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.603·11-s + 1.66·13-s − 0.970·17-s − 0.917·19-s + 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.328·37-s + 0.609·43-s − 1.75·47-s + 0.824·53-s + 1.07·55-s + 1.04·59-s − 0.768·61-s + 2.97·65-s + 0.977·67-s + 1.66·71-s + 0.234·73-s − 1.35·79-s + 0.439·83-s − 1.73·85-s − 1.64·95-s + 0.203·97-s + 1.59·101-s − 1.57·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.431775895\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.431775895\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.188009173971352962353522111319, −6.84547186300605263895597001246, −6.47622046568330368740547810122, −6.00007481606225187768786369361, −5.24733381494387031366902608811, −4.40221900295040807779267151750, −3.55396593058750079230795234813, −2.51518059723836198260317180881, −1.81321664819287647110164631158, −1.01202132858862645131480480035,
1.01202132858862645131480480035, 1.81321664819287647110164631158, 2.51518059723836198260317180881, 3.55396593058750079230795234813, 4.40221900295040807779267151750, 5.24733381494387031366902608811, 6.00007481606225187768786369361, 6.47622046568330368740547810122, 6.84547186300605263895597001246, 8.188009173971352962353522111319