| L(s) = 1 | − 3-s − 7-s + 9-s − 5·11-s − 3·13-s + 2·17-s − 3·19-s + 21-s − 23-s − 27-s + 29-s + 5·33-s − 4·37-s + 3·39-s − 3·41-s + 5·43-s + 6·47-s − 6·49-s − 2·51-s + 3·57-s − 6·59-s − 4·61-s − 63-s − 4·67-s + 69-s + 2·71-s + 9·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.832·13-s + 0.485·17-s − 0.688·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.870·33-s − 0.657·37-s + 0.480·39-s − 0.468·41-s + 0.762·43-s + 0.875·47-s − 6/7·49-s − 0.280·51-s + 0.397·57-s − 0.781·59-s − 0.512·61-s − 0.125·63-s − 0.488·67-s + 0.120·69-s + 0.237·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.71383548589763, −13.46494074719089, −12.73177334065681, −12.42463393416932, −12.17266978555289, −11.44024879520024, −10.80431250621001, −10.55532336774590, −10.09962454516946, −9.547899819427508, −9.120870743709920, −8.226276897519937, −8.009065232395308, −7.358237289091343, −6.921556539672962, −6.318521591483334, −5.714343629385299, −5.319050867806767, −4.768745104942706, −4.277403411199392, −3.474321159464384, −2.871371608377101, −2.319749550327965, −1.646121194180136, −0.6124382122820680, 0,
0.6124382122820680, 1.646121194180136, 2.319749550327965, 2.871371608377101, 3.474321159464384, 4.277403411199392, 4.768745104942706, 5.319050867806767, 5.714343629385299, 6.318521591483334, 6.921556539672962, 7.358237289091343, 8.009065232395308, 8.226276897519937, 9.120870743709920, 9.547899819427508, 10.09962454516946, 10.55532336774590, 10.80431250621001, 11.44024879520024, 12.17266978555289, 12.42463393416932, 12.73177334065681, 13.46494074719089, 13.71383548589763
