| L(s) = 1 | + 3.10·3-s + 0.813·5-s − 0.813·7-s + 6.62·9-s − 1.71·11-s + 6.20·13-s + 2.52·15-s − 5.91·17-s − 19-s − 2.52·21-s + 4·23-s − 4.33·25-s + 11.2·27-s + 3.68·29-s + 10.3·31-s − 5.30·33-s − 0.661·35-s − 2.52·37-s + 19.2·39-s + 1.68·41-s + 3.33·43-s + 5.39·45-s − 5.86·47-s − 6.33·49-s − 18.3·51-s + 5.94·53-s − 1.39·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 0.363·5-s − 0.307·7-s + 2.20·9-s − 0.515·11-s + 1.72·13-s + 0.651·15-s − 1.43·17-s − 0.229·19-s − 0.550·21-s + 0.834·23-s − 0.867·25-s + 2.16·27-s + 0.683·29-s + 1.86·31-s − 0.924·33-s − 0.111·35-s − 0.415·37-s + 3.08·39-s + 0.262·41-s + 0.509·43-s + 0.803·45-s − 0.855·47-s − 0.905·49-s − 2.57·51-s + 0.816·53-s − 0.187·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.354352089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.354352089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 0.813T + 5T^{2} \) |
| 7 | \( 1 + 0.813T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 3.68T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 3.33T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 8.64T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 6.15T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.459988631250498286567100752978, −7.86354332437703994414342525521, −6.70735852721148669951675757316, −6.49470290498488940317184982866, −5.23793277569753189864460420820, −4.21545955401536147414142872139, −3.67839275411268903435773525884, −2.75555570049778777513910216076, −2.21145624622053697853736077715, −1.12474751477503475778201019196,
1.12474751477503475778201019196, 2.21145624622053697853736077715, 2.75555570049778777513910216076, 3.67839275411268903435773525884, 4.21545955401536147414142872139, 5.23793277569753189864460420820, 6.49470290498488940317184982866, 6.70735852721148669951675757316, 7.86354332437703994414342525521, 8.459988631250498286567100752978
