| L(s) = 1 | − 2·3-s + 5-s + 9-s + 3·11-s − 5·13-s − 2·15-s + 6·17-s − 19-s − 3·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s − 6·33-s − 11·37-s + 10·39-s + 3·41-s − 10·43-s + 45-s − 3·47-s − 12·51-s − 3·53-s + 3·55-s + 2·57-s + 4·61-s − 5·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.38·13-s − 0.516·15-s + 1.45·17-s − 0.229·19-s − 0.625·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.04·33-s − 1.80·37-s + 1.60·39-s + 0.468·41-s − 1.52·43-s + 0.149·45-s − 0.437·47-s − 1.68·51-s − 0.412·53-s + 0.404·55-s + 0.264·57-s + 0.512·61-s − 0.620·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.34755569981787, −15.99852193001746, −15.07849731160264, −14.51767287688415, −14.17121810958494, −13.56873195431973, −12.63157966239224, −12.20146268896425, −11.88480101959004, −11.42943581962299, −10.43266977821791, −10.14801942173579, −9.735721051352669, −8.843610733299129, −8.274679932556058, −7.414013167148964, −6.854547010441639, −6.223299008979212, −5.770909219907725, −4.927060831793643, −4.731750005131754, −3.593517906534936, −2.871979854685591, −1.856951675032093, −1.029387530647764, 0,
1.029387530647764, 1.856951675032093, 2.871979854685591, 3.593517906534936, 4.731750005131754, 4.927060831793643, 5.770909219907725, 6.223299008979212, 6.854547010441639, 7.414013167148964, 8.274679932556058, 8.843610733299129, 9.735721051352669, 10.14801942173579, 10.43266977821791, 11.42943581962299, 11.88480101959004, 12.20146268896425, 12.63157966239224, 13.56873195431973, 14.17121810958494, 14.51767287688415, 15.07849731160264, 15.99852193001746, 16.34755569981787
