| L(s) = 1 | + 2.30·2-s − 1.30·3-s + 3.30·4-s − 3·6-s + 4.30·7-s + 3.00·8-s − 1.30·9-s − 11-s − 4.30·12-s + 5·13-s + 9.90·14-s + 0.302·16-s − 3.90·17-s − 3.00·18-s − 19-s − 5.60·21-s − 2.30·22-s − 3.69·23-s − 3.90·24-s + 11.5·26-s + 5.60·27-s + 14.2·28-s − 9.90·29-s − 4.21·31-s − 5.30·32-s + 1.30·33-s − 9·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.752·3-s + 1.65·4-s − 1.22·6-s + 1.62·7-s + 1.06·8-s − 0.434·9-s − 0.301·11-s − 1.24·12-s + 1.38·13-s + 2.64·14-s + 0.0756·16-s − 0.947·17-s − 0.707·18-s − 0.229·19-s − 1.22·21-s − 0.490·22-s − 0.770·23-s − 0.797·24-s + 2.25·26-s + 1.07·27-s + 2.68·28-s − 1.83·29-s − 0.756·31-s − 0.937·32-s + 0.226·33-s − 1.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543835146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543835146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.30T + 3T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 0.0916T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 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
−11.75264289167998397713824240051, −11.24933123400379278318604389692, −10.83460400463024479003249245451, −8.843291587354579709770843927933, −7.78167051977930937994868351532, −6.36993791897891590419911293865, −5.62137348147238836438259296365, −4.80175039695978728299343645479, −3.79894828953799276588994147306, −2.04290806640930588735681357089,
2.04290806640930588735681357089, 3.79894828953799276588994147306, 4.80175039695978728299343645479, 5.62137348147238836438259296365, 6.36993791897891590419911293865, 7.78167051977930937994868351532, 8.843291587354579709770843927933, 10.83460400463024479003249245451, 11.24933123400379278318604389692, 11.75264289167998397713824240051
