| L(s) = 1 | + 1.11·2-s − 2.75·3-s − 0.755·4-s − 1.34·5-s − 3.07·6-s − 3.07·8-s + 4.58·9-s − 1.49·10-s − 1.07·11-s + 2.08·12-s − 0.966·13-s + 3.70·15-s − 1.91·16-s − 6.52·17-s + 5.11·18-s − 7.07·19-s + 1.01·20-s − 1.20·22-s + 0.898·23-s + 8.46·24-s − 3.19·25-s − 1.07·26-s − 4.36·27-s + 2.82·29-s + 4.13·30-s − 1.38·31-s + 4.00·32-s + ⋯ |
| L(s) = 1 | + 0.788·2-s − 1.59·3-s − 0.377·4-s − 0.601·5-s − 1.25·6-s − 1.08·8-s + 1.52·9-s − 0.474·10-s − 0.324·11-s + 0.600·12-s − 0.267·13-s + 0.956·15-s − 0.479·16-s − 1.58·17-s + 1.20·18-s − 1.62·19-s + 0.227·20-s − 0.256·22-s + 0.187·23-s + 1.72·24-s − 0.638·25-s − 0.211·26-s − 0.840·27-s + 0.524·29-s + 0.754·30-s − 0.249·31-s + 0.708·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3878450095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3878450095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 + 0.966T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 - 0.898T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 9.30T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−9.083522833444671572227123384863, −8.447994156949966578514526278969, −7.26345245990590482752751556676, −6.47072717379555788176042356083, −5.91130058401932565622188805452, −4.98543454329405657581301618385, −4.47948244254060999838976710025, −3.78938735684821745655208980763, −2.29547828643854049240363008482, −0.37872339195969251892918295418,
0.37872339195969251892918295418, 2.29547828643854049240363008482, 3.78938735684821745655208980763, 4.47948244254060999838976710025, 4.98543454329405657581301618385, 5.91130058401932565622188805452, 6.47072717379555788176042356083, 7.26345245990590482752751556676, 8.447994156949966578514526278969, 9.083522833444671572227123384863
