| L(s) = 1 | − 10.2·3-s + 101.·7-s − 138.·9-s + 333.·11-s + 25.3·13-s − 1.69e3·17-s − 1.95e3·19-s − 1.03e3·21-s + 2.85e3·23-s + 3.90e3·27-s + 6.62e3·29-s + 7.47e3·31-s − 3.40e3·33-s − 1.52e4·37-s − 259.·39-s + 1.06e3·41-s − 4.27e3·43-s + 2.06e3·47-s − 6.53e3·49-s + 1.73e4·51-s − 4.91e3·53-s + 1.99e4·57-s − 4.70e3·59-s + 4.35e4·61-s − 1.40e4·63-s − 5.81e4·67-s − 2.91e4·69-s + ⋯ |
| L(s) = 1 | − 0.655·3-s + 0.781·7-s − 0.569·9-s + 0.830·11-s + 0.0416·13-s − 1.42·17-s − 1.24·19-s − 0.512·21-s + 1.12·23-s + 1.02·27-s + 1.46·29-s + 1.39·31-s − 0.544·33-s − 1.83·37-s − 0.0273·39-s + 0.0986·41-s − 0.352·43-s + 0.136·47-s − 0.388·49-s + 0.934·51-s − 0.240·53-s + 0.815·57-s − 0.176·59-s + 1.49·61-s − 0.445·63-s − 1.58·67-s − 0.736·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.2T + 243T^{2} \) |
| 7 | \( 1 - 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 333.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 25.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.69e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.27e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.91e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.70e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.68e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 204.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.38e4T + 8.58e9T^{2} \) |
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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
−10.26163084883998151195819758215, −8.758856355612628905389018472591, −8.493802558724932454495306590465, −6.87752155248816714770188898049, −6.30953556114698836569593234703, −5.03617701085424120100931087369, −4.31051160216046930565057202717, −2.71156408631087238510925474619, −1.35256567497541924694857971703, 0,
1.35256567497541924694857971703, 2.71156408631087238510925474619, 4.31051160216046930565057202717, 5.03617701085424120100931087369, 6.30953556114698836569593234703, 6.87752155248816714770188898049, 8.493802558724932454495306590465, 8.758856355612628905389018472591, 10.26163084883998151195819758215
