| L(s) = 1 | + 2.40·3-s − 4.41·7-s + 2.76·9-s − 2.29·11-s + 6.92·13-s + 1.51·17-s − 2.89·19-s − 10.5·21-s − 23-s − 0.570·27-s − 7.68·29-s − 3.85·31-s − 5.50·33-s + 8.62·37-s + 16.6·39-s − 6.44·41-s + 3.48·43-s + 6.19·47-s + 12.4·49-s + 3.63·51-s + 2.17·53-s − 6.95·57-s + 11.7·59-s − 5.11·61-s − 12.1·63-s − 9.94·67-s − 2.40·69-s + ⋯ |
| L(s) = 1 | + 1.38·3-s − 1.66·7-s + 0.920·9-s − 0.691·11-s + 1.92·13-s + 0.367·17-s − 0.665·19-s − 2.31·21-s − 0.208·23-s − 0.109·27-s − 1.42·29-s − 0.692·31-s − 0.958·33-s + 1.41·37-s + 2.66·39-s − 1.00·41-s + 0.531·43-s + 0.903·47-s + 1.78·49-s + 0.508·51-s + 0.299·53-s − 0.921·57-s + 1.53·59-s − 0.654·61-s − 1.53·63-s − 1.21·67-s − 0.288·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.62T + 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 + 9.94T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 + 8.04T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.50181606353906745017633154404, −6.77904577168746900117172698042, −5.98119948619663594923359414121, −5.60304267731364295174122109762, −4.01654281754674569415451836006, −3.81393877548382263420594682735, −3.02948467624602344359305953154, −2.48124212432530265462058395912, −1.40815048300410945079438049643, 0,
1.40815048300410945079438049643, 2.48124212432530265462058395912, 3.02948467624602344359305953154, 3.81393877548382263420594682735, 4.01654281754674569415451836006, 5.60304267731364295174122109762, 5.98119948619663594923359414121, 6.77904577168746900117172698042, 7.50181606353906745017633154404
