| L(s) = 1 | + 3-s − 1.56·5-s − 7-s + 9-s + 2·11-s − 13-s − 1.56·15-s + 5.12·17-s + 2.43·19-s − 21-s − 4.68·23-s − 2.56·25-s + 27-s − 3.56·29-s − 1.56·31-s + 2·33-s + 1.56·35-s + 1.12·37-s − 39-s + 7.12·41-s + 9.56·43-s − 1.56·45-s + 6.68·47-s + 49-s + 5.12·51-s + 0.438·53-s − 3.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.698·5-s − 0.377·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s − 0.403·15-s + 1.24·17-s + 0.559·19-s − 0.218·21-s − 0.976·23-s − 0.512·25-s + 0.192·27-s − 0.661·29-s − 0.280·31-s + 0.348·33-s + 0.263·35-s + 0.184·37-s − 0.160·39-s + 1.11·41-s + 1.45·43-s − 0.232·45-s + 0.975·47-s + 0.142·49-s + 0.717·51-s + 0.0602·53-s − 0.421·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008011801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008011801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 - 5.56T + 89T^{2} \) |
| 97 | \( 1 + 7.56T + 97T^{2} \) |
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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
−8.209197905554803173198414239762, −7.63440507596170888808173101697, −7.22429220934273708928997619655, −6.12296435506030564500514203359, −5.52373118515310058780871164410, −4.32082479291631640997077825386, −3.79083645770729656009810710698, −3.06498976679320005147287416152, −2.01780281347619080587613979567, −0.77977158341817037127408942396,
0.77977158341817037127408942396, 2.01780281347619080587613979567, 3.06498976679320005147287416152, 3.79083645770729656009810710698, 4.32082479291631640997077825386, 5.52373118515310058780871164410, 6.12296435506030564500514203359, 7.22429220934273708928997619655, 7.63440507596170888808173101697, 8.209197905554803173198414239762
