| L(s) = 1 | − 1.89·2-s − 2.50·3-s + 1.59·4-s + 5-s + 4.74·6-s + 7-s + 0.762·8-s + 3.26·9-s − 1.89·10-s − 3.56·11-s − 3.99·12-s − 1.73·13-s − 1.89·14-s − 2.50·15-s − 4.64·16-s − 1.29·17-s − 6.18·18-s − 3.92·19-s + 1.59·20-s − 2.50·21-s + 6.76·22-s − 3.79·23-s − 1.90·24-s + 25-s + 3.28·26-s − 0.657·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 1.44·3-s + 0.798·4-s + 0.447·5-s + 1.93·6-s + 0.377·7-s + 0.269·8-s + 1.08·9-s − 0.599·10-s − 1.07·11-s − 1.15·12-s − 0.480·13-s − 0.506·14-s − 0.646·15-s − 1.16·16-s − 0.314·17-s − 1.45·18-s − 0.901·19-s + 0.357·20-s − 0.546·21-s + 1.44·22-s − 0.790·23-s − 0.389·24-s + 0.200·25-s + 0.644·26-s − 0.126·27-s + 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 + 5.90T + 43T^{2} \) |
| 47 | \( 1 - 0.778T + 47T^{2} \) |
| 53 | \( 1 + 1.18T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 4.64T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 + 1.00T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.58559952693636932530120084070, −6.75785800122488430108657284362, −6.31455858896179036156023986396, −5.43182917039499005095538635466, −4.86567422016524800098391360730, −4.24426008078566551881131525772, −2.63207989070175742040683278240, −1.90765978964222249221661410214, −0.834274511284980009731570224972, 0,
0.834274511284980009731570224972, 1.90765978964222249221661410214, 2.63207989070175742040683278240, 4.24426008078566551881131525772, 4.86567422016524800098391360730, 5.43182917039499005095538635466, 6.31455858896179036156023986396, 6.75785800122488430108657284362, 7.58559952693636932530120084070
