| L(s) = 1 | − 1.40·3-s + 3.61·5-s − 1.02·9-s − 5.64·11-s + 6.54·13-s − 5.07·15-s + 3.39·17-s + 5.19·19-s − 2.81·23-s + 8.07·25-s + 5.65·27-s + 2.17·29-s + 6.26·31-s + 7.93·33-s + 0.641·37-s − 9.18·39-s − 41-s − 4.44·43-s − 3.71·45-s + 6.69·47-s − 4.76·51-s + 4.95·53-s − 20.4·55-s − 7.29·57-s − 11.0·59-s − 12.0·61-s + 23.6·65-s + ⋯ |
| L(s) = 1 | − 0.810·3-s + 1.61·5-s − 0.342·9-s − 1.70·11-s + 1.81·13-s − 1.31·15-s + 0.822·17-s + 1.19·19-s − 0.585·23-s + 1.61·25-s + 1.08·27-s + 0.404·29-s + 1.12·31-s + 1.38·33-s + 0.105·37-s − 1.47·39-s − 0.156·41-s − 0.677·43-s − 0.554·45-s + 0.976·47-s − 0.666·51-s + 0.680·53-s − 2.75·55-s − 0.966·57-s − 1.43·59-s − 1.54·61-s + 2.93·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.139210754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.139210754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 2.81T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 0.641T + 37T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 0.422T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.944070383571078156748406405420, −6.94106820446540225465111997906, −6.03751050374712395167890023791, −5.84114106974453062059595334755, −5.34924068210010641464768801793, −4.60848407723921746480185649647, −3.23465588350775488703715475542, −2.73302996545315855080368529332, −1.62849833859940972570829587107, −0.790362379969394607797797583562,
0.790362379969394607797797583562, 1.62849833859940972570829587107, 2.73302996545315855080368529332, 3.23465588350775488703715475542, 4.60848407723921746480185649647, 5.34924068210010641464768801793, 5.84114106974453062059595334755, 6.03751050374712395167890023791, 6.94106820446540225465111997906, 7.944070383571078156748406405420
