| L(s) = 1 | + 0.732·3-s + 2.44·5-s − 3.86·7-s − 2.46·9-s − 4.73·11-s + 0.378·13-s + 1.79·15-s − 3.46·17-s − 4.73·19-s − 2.82·21-s − 1.79·23-s + 0.999·25-s − 4·27-s − 2.44·29-s + 5.65·31-s − 3.46·33-s − 9.46·35-s − 5.27·37-s + 0.277·39-s + 6.92·41-s + 2.19·43-s − 6.03·45-s + 9.79·47-s + 7.92·49-s − 2.53·51-s + 10.9·53-s − 11.5·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 1.09·5-s − 1.46·7-s − 0.821·9-s − 1.42·11-s + 0.105·13-s + 0.462·15-s − 0.840·17-s − 1.08·19-s − 0.617·21-s − 0.373·23-s + 0.199·25-s − 0.769·27-s − 0.454·29-s + 1.01·31-s − 0.603·33-s − 1.59·35-s − 0.867·37-s + 0.0444·39-s + 1.08·41-s + 0.334·43-s − 0.899·45-s + 1.42·47-s + 1.13·49-s − 0.355·51-s + 1.50·53-s − 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.378T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 - 5.27T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−9.459329416778827821160423516659, −8.882934653974771379726078210214, −7.999101773923296326052391324414, −6.84850762097936818446039271828, −6.02438593912573896226785690297, −5.49309552131070252349883095021, −4.06104475041743640948550770173, −2.76767903625051041888723287646, −2.32656645388422014905720401479, 0,
2.32656645388422014905720401479, 2.76767903625051041888723287646, 4.06104475041743640948550770173, 5.49309552131070252349883095021, 6.02438593912573896226785690297, 6.84850762097936818446039271828, 7.999101773923296326052391324414, 8.882934653974771379726078210214, 9.459329416778827821160423516659
