L(s) = 1 | + 1.41·3-s + 5-s + 4.24·7-s − 0.999·9-s + 5.65·11-s − 2·13-s + 1.41·15-s + 6·17-s − 2.82·19-s + 6·21-s − 7.07·23-s + 25-s − 5.65·27-s − 4·29-s − 2.82·31-s + 8.00·33-s + 4.24·35-s + 2·37-s − 2.82·39-s + 8·41-s − 1.41·43-s − 0.999·45-s − 1.41·47-s + 10.9·49-s + 8.48·51-s − 2·53-s + 5.65·55-s + ⋯ |
L(s) = 1 | + 0.816·3-s + 0.447·5-s + 1.60·7-s − 0.333·9-s + 1.70·11-s − 0.554·13-s + 0.365·15-s + 1.45·17-s − 0.648·19-s + 1.30·21-s − 1.47·23-s + 0.200·25-s − 1.08·27-s − 0.742·29-s − 0.508·31-s + 1.39·33-s + 0.717·35-s + 0.328·37-s − 0.452·39-s + 1.24·41-s − 0.215·43-s − 0.149·45-s − 0.206·47-s + 1.57·49-s + 1.18·51-s − 0.274·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.866495792\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.866495792\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.417813960759149650388470402767, −8.934543866025143240598883397938, −7.965800338953623128029847401656, −7.60923321559064389382275585544, −6.26415304508219311984899254644, −5.49936600934448183561518176378, −4.40058944195331033734735040382, −3.58082100235383699110499102578, −2.22259305792153845108702480225, −1.45358908317161389346517381853,
1.45358908317161389346517381853, 2.22259305792153845108702480225, 3.58082100235383699110499102578, 4.40058944195331033734735040382, 5.49936600934448183561518176378, 6.26415304508219311984899254644, 7.60923321559064389382275585544, 7.965800338953623128029847401656, 8.934543866025143240598883397938, 9.417813960759149650388470402767