| L(s) = 1 | + 0.0723·2-s − 3.29·3-s − 1.99·4-s + 2.88·5-s − 0.238·6-s + 1.08·7-s − 0.289·8-s + 7.82·9-s + 0.208·10-s + 0.496·11-s + 6.56·12-s + 0.0782·14-s − 9.49·15-s + 3.96·16-s − 3.56·17-s + 0.566·18-s − 0.808·19-s − 5.75·20-s − 3.55·21-s + 0.0359·22-s + 7.31·23-s + 0.951·24-s + 3.32·25-s − 15.8·27-s − 2.15·28-s − 3.67·29-s − 0.687·30-s + ⋯ |
| L(s) = 1 | + 0.0511·2-s − 1.89·3-s − 0.997·4-s + 1.29·5-s − 0.0972·6-s + 0.408·7-s − 0.102·8-s + 2.60·9-s + 0.0660·10-s + 0.149·11-s + 1.89·12-s + 0.0209·14-s − 2.45·15-s + 0.992·16-s − 0.863·17-s + 0.133·18-s − 0.185·19-s − 1.28·20-s − 0.776·21-s + 0.00766·22-s + 1.52·23-s + 0.194·24-s + 0.665·25-s − 3.05·27-s − 0.407·28-s − 0.682·29-s − 0.125·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0723T + 2T^{2} \) |
| 3 | \( 1 + 3.29T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 0.496T + 11T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 0.808T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.69301720579075867477859500124, −6.70723252805623865276711628942, −6.37101343302259934404268128764, −5.39649525110730688911198136917, −5.16456605968335212494077417083, −4.57169820103481884606747559248, −3.57097418477504518968466329467, −1.94737327608702667325598902442, −1.16352955499703174897929188661, 0,
1.16352955499703174897929188661, 1.94737327608702667325598902442, 3.57097418477504518968466329467, 4.57169820103481884606747559248, 5.16456605968335212494077417083, 5.39649525110730688911198136917, 6.37101343302259934404268128764, 6.70723252805623865276711628942, 7.69301720579075867477859500124
