| L(s) = 1 | + 2·3-s + 3.46·5-s − 1.26·7-s + 9-s + 4.73·11-s − 13-s + 6.92·15-s + 5.46·17-s + 0.732·19-s − 2.53·21-s + 4·23-s + 6.99·25-s − 4·27-s − 2·29-s + 6.73·31-s + 9.46·33-s − 4.39·35-s − 8.92·37-s − 2·39-s − 8.92·41-s + 0.535·43-s + 3.46·45-s − 6.73·47-s − 5.39·49-s + 10.9·51-s − 2.92·53-s + 16.3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.54·5-s − 0.479·7-s + 0.333·9-s + 1.42·11-s − 0.277·13-s + 1.78·15-s + 1.32·17-s + 0.167·19-s − 0.553·21-s + 0.834·23-s + 1.39·25-s − 0.769·27-s − 0.371·29-s + 1.20·31-s + 1.64·33-s − 0.742·35-s − 1.46·37-s − 0.320·39-s − 1.39·41-s + 0.0817·43-s + 0.516·45-s − 0.981·47-s − 0.770·49-s + 1.53·51-s − 0.402·53-s + 2.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.017906182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.017906182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 0.535T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 - 0.732T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 - 3.26T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.816063918357750841965306242757, −8.076379003824851562279712457653, −7.02211700383963993985124750568, −6.46619281447543899537810230975, −5.66889385018085002832535085232, −4.87019189948839514775394256277, −3.53119595213668869765394439984, −3.10910910375638935604081958113, −2.04576591524775825099739875786, −1.28887760380532648330649426907,
1.28887760380532648330649426907, 2.04576591524775825099739875786, 3.10910910375638935604081958113, 3.53119595213668869765394439984, 4.87019189948839514775394256277, 5.66889385018085002832535085232, 6.46619281447543899537810230975, 7.02211700383963993985124750568, 8.076379003824851562279712457653, 8.816063918357750841965306242757
