| L(s) = 1 | + 2.17·2-s + 1.70·3-s + 2.70·4-s − 5-s + 3.70·6-s + 1.53·8-s − 0.0783·9-s − 2.17·10-s − 0.630·11-s + 4.63·12-s + 4.34·13-s − 1.70·15-s − 2.07·16-s + 1.55·17-s − 0.170·18-s + 5.70·19-s − 2.70·20-s − 1.36·22-s + 6.63·23-s + 2.63·24-s + 25-s + 9.41·26-s − 5.26·27-s − 29-s − 3.70·30-s + 2.29·31-s − 7.58·32-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.986·3-s + 1.35·4-s − 0.447·5-s + 1.51·6-s + 0.544·8-s − 0.0261·9-s − 0.686·10-s − 0.190·11-s + 1.33·12-s + 1.20·13-s − 0.441·15-s − 0.519·16-s + 0.376·17-s − 0.0400·18-s + 1.30·19-s − 0.605·20-s − 0.291·22-s + 1.38·23-s + 0.537·24-s + 0.200·25-s + 1.84·26-s − 1.01·27-s − 0.185·29-s − 0.677·30-s + 0.411·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.867820707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.867820707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 11 | \( 1 + 0.630T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 - 0.921T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.896893150837476291374593327448, −7.09233868139041141591629445526, −6.49594097689542842098800646270, −5.43688721272347723904718830082, −5.24922504875110803954831920920, −4.06462994948761441162295704627, −3.59810000857623001231369606742, −3.04374273963547748302740249037, −2.35173584175881314080772769128, −1.04597890059242463270513278043,
1.04597890059242463270513278043, 2.35173584175881314080772769128, 3.04374273963547748302740249037, 3.59810000857623001231369606742, 4.06462994948761441162295704627, 5.24922504875110803954831920920, 5.43688721272347723904718830082, 6.49594097689542842098800646270, 7.09233868139041141591629445526, 7.896893150837476291374593327448
