| L(s) = 1 | + 1.34·2-s − 2.80·3-s − 0.183·4-s − 2.23·5-s − 3.77·6-s − 2.93·7-s − 2.94·8-s + 4.85·9-s − 3.00·10-s + 3.16·11-s + 0.513·12-s + 4.16·13-s − 3.95·14-s + 6.25·15-s − 3.60·16-s − 4.55·17-s + 6.54·18-s − 6.97·19-s + 0.408·20-s + 8.22·21-s + 4.26·22-s + 5.32·23-s + 8.24·24-s − 0.0157·25-s + 5.61·26-s − 5.20·27-s + 0.537·28-s + ⋯ |
| L(s) = 1 | + 0.953·2-s − 1.61·3-s − 0.0915·4-s − 0.998·5-s − 1.54·6-s − 1.10·7-s − 1.04·8-s + 1.61·9-s − 0.951·10-s + 0.953·11-s + 0.148·12-s + 1.15·13-s − 1.05·14-s + 1.61·15-s − 0.900·16-s − 1.10·17-s + 1.54·18-s − 1.59·19-s + 0.0914·20-s + 1.79·21-s + 0.908·22-s + 1.11·23-s + 1.68·24-s − 0.00315·25-s + 1.10·26-s − 1.00·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03079713477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03079713477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
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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
−7.18199994780286504537331751695, −6.68144151116724544001654474999, −6.27300291153198540347583467826, −5.69932114112062599274846601166, −4.90863345634163672292425840479, −4.05218859550791781855504594704, −3.96079573765343031046521815888, −3.06305781728854255340867885182, −1.52720305303649668363537369401, −0.080280035324800767044311496380,
0.080280035324800767044311496380, 1.52720305303649668363537369401, 3.06305781728854255340867885182, 3.96079573765343031046521815888, 4.05218859550791781855504594704, 4.90863345634163672292425840479, 5.69932114112062599274846601166, 6.27300291153198540347583467826, 6.68144151116724544001654474999, 7.18199994780286504537331751695
