| L(s) = 1 | − 2.31e3·2-s + 4.50e4·3-s + 3.26e6·4-s + 9.76e6·5-s − 1.04e8·6-s − 6.93e8·7-s − 2.70e9·8-s − 8.42e9·9-s − 2.26e10·10-s − 1.34e10·11-s + 1.47e11·12-s + 7.82e11·13-s + 1.60e12·14-s + 4.40e11·15-s − 5.82e11·16-s − 3.22e12·17-s + 1.95e13·18-s + 9.87e12·19-s + 3.18e13·20-s − 3.12e13·21-s + 3.12e13·22-s + 2.96e14·23-s − 1.21e14·24-s + 9.53e13·25-s − 1.81e15·26-s − 8.51e14·27-s − 2.26e15·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.440·3-s + 1.55·4-s + 0.447·5-s − 0.704·6-s − 0.928·7-s − 0.890·8-s − 0.805·9-s − 0.715·10-s − 0.156·11-s + 0.686·12-s + 1.57·13-s + 1.48·14-s + 0.197·15-s − 0.132·16-s − 0.387·17-s + 1.28·18-s + 0.369·19-s + 0.696·20-s − 0.408·21-s + 0.250·22-s + 1.49·23-s − 0.392·24-s + 0.199·25-s − 2.51·26-s − 0.795·27-s − 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.8955539972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8955539972\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 9.76e6T \) |
| good | 2 | \( 1 + 2.31e3T + 2.09e6T^{2} \) |
| 3 | \( 1 - 4.50e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 6.93e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.34e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.82e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 3.22e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 9.87e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.96e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.05e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.62e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.49e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.44e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.52e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 7.91e15T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.16e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.98e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 2.24e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 5.10e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.66e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.85e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 8.75e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.76e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.17e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.33e20T + 5.27e41T^{2} \) |
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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
−18.44595560985214180575435253423, −17.08020132059963972356146290196, −15.75389036167753911631255249774, −13.50735931277945169483163522672, −11.01990662589309415313501850955, −9.471507612001995553743950202520, −8.388553339304546418356830675414, −6.42189344282629746367641287488, −2.82049632807706720445878209404, −0.894784747208873150369404162907,
0.894784747208873150369404162907, 2.82049632807706720445878209404, 6.42189344282629746367641287488, 8.388553339304546418356830675414, 9.471507612001995553743950202520, 11.01990662589309415313501850955, 13.50735931277945169483163522672, 15.75389036167753911631255249774, 17.08020132059963972356146290196, 18.44595560985214180575435253423
