| L(s) = 1 | − 5-s − 2·7-s + 3·11-s + 5·13-s − 3·17-s + 7·23-s + 25-s + 5·29-s + 5·31-s + 2·35-s + 6·37-s − 9·43-s + 9·47-s − 3·49-s − 8·53-s − 3·55-s − 4·59-s − 14·61-s − 5·65-s − 12·67-s − 8·71-s − 2·73-s − 6·77-s + 11·79-s + 6·83-s + 3·85-s − 10·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s + 1.38·13-s − 0.727·17-s + 1.45·23-s + 1/5·25-s + 0.928·29-s + 0.898·31-s + 0.338·35-s + 0.986·37-s − 1.37·43-s + 1.31·47-s − 3/7·49-s − 1.09·53-s − 0.404·55-s − 0.520·59-s − 1.79·61-s − 0.620·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.683·77-s + 1.23·79-s + 0.658·83-s + 0.325·85-s − 1.04·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924576246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924576246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.77881102158928034253035114667, −6.93642925585380834215807660775, −6.37467773759916669255718466348, −6.01886913977402411294715913543, −4.75784261927739767646854894252, −4.30592556212690991803739153586, −3.33853366148878817312601693981, −2.95692333731232252605376648215, −1.58030918878095248766663987309, −0.71528966973463294278594770067,
0.71528966973463294278594770067, 1.58030918878095248766663987309, 2.95692333731232252605376648215, 3.33853366148878817312601693981, 4.30592556212690991803739153586, 4.75784261927739767646854894252, 6.01886913977402411294715913543, 6.37467773759916669255718466348, 6.93642925585380834215807660775, 7.77881102158928034253035114667
