| L(s) = 1 | − 3.22·5-s + 2.37·7-s + 2.20·11-s + 2·13-s − 3.22·17-s − 19-s − 1.01·23-s + 5.37·25-s + 1.01·29-s − 4.74·31-s − 7.64·35-s + 10.7·37-s − 5.43·41-s + 11.1·43-s + 4.23·47-s − 1.37·49-s + 9.84·53-s − 7.11·55-s − 10.8·59-s − 5.11·61-s − 6.44·65-s + 4·67-s − 2.02·71-s − 5.11·73-s + 5.24·77-s + 4·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s + 0.896·7-s + 0.666·11-s + 0.554·13-s − 0.781·17-s − 0.229·19-s − 0.210·23-s + 1.07·25-s + 0.187·29-s − 0.852·31-s − 1.29·35-s + 1.76·37-s − 0.848·41-s + 1.69·43-s + 0.617·47-s − 0.196·49-s + 1.35·53-s − 0.959·55-s − 1.41·59-s − 0.655·61-s − 0.798·65-s + 0.488·67-s − 0.239·71-s − 0.598·73-s + 0.597·77-s + 0.450·79-s + 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483444259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483444259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2.02T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−8.769359712107632604034576610804, −7.958604350175653133854217056348, −7.53751750734148443060072564470, −6.63663558871069114094398036591, −5.77308214461059582812985593409, −4.56426651941744798108376417970, −4.20762262891916088886630148617, −3.33642056655243582922104858778, −2.04243174127748300706522723755, −0.77047052557336047411496168979,
0.77047052557336047411496168979, 2.04243174127748300706522723755, 3.33642056655243582922104858778, 4.20762262891916088886630148617, 4.56426651941744798108376417970, 5.77308214461059582812985593409, 6.63663558871069114094398036591, 7.53751750734148443060072564470, 7.958604350175653133854217056348, 8.769359712107632604034576610804
