| L(s) = 1 | + 2-s − 1.41·3-s − 4-s − 1.41·6-s + 4.24·7-s − 3·8-s − 0.999·9-s + 4.24·11-s + 1.41·12-s + 4.24·14-s − 16-s − 0.999·18-s − 6·19-s − 6·21-s + 4.24·22-s − 1.41·23-s + 4.24·24-s + 5.65·27-s − 4.24·28-s + 4.24·29-s + 1.41·31-s + 5·32-s − 6·33-s + 0.999·36-s − 4.24·37-s − 6·38-s − 4.24·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.816·3-s − 0.5·4-s − 0.577·6-s + 1.60·7-s − 1.06·8-s − 0.333·9-s + 1.27·11-s + 0.408·12-s + 1.13·14-s − 0.250·16-s − 0.235·18-s − 1.37·19-s − 1.30·21-s + 0.904·22-s − 0.294·23-s + 0.866·24-s + 1.08·27-s − 0.801·28-s + 0.787·29-s + 0.254·31-s + 0.883·32-s − 1.04·33-s + 0.166·36-s − 0.697·37-s − 0.973·38-s − 0.662·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.950714695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.950714695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 9.89T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.034644587788928071412803740751, −6.99366954175650791454059751724, −6.25976830874351044227757554532, −5.71784662991099448350004152575, −5.04440987493722448484406740064, −4.36258996367297469945829769524, −4.03633826295134161834150902256, −2.80616085018079810668167867651, −1.72379836756367044463261245269, −0.68700246874957017145831642276,
0.68700246874957017145831642276, 1.72379836756367044463261245269, 2.80616085018079810668167867651, 4.03633826295134161834150902256, 4.36258996367297469945829769524, 5.04440987493722448484406740064, 5.71784662991099448350004152575, 6.25976830874351044227757554532, 6.99366954175650791454059751724, 8.034644587788928071412803740751
