| L(s) = 1 | + 21.3·2-s + 195.·3-s − 57.5·4-s + 4.15e3·6-s + 2.27e3·7-s − 1.21e4·8-s + 1.83e4·9-s − 7.17e3·11-s − 1.12e4·12-s − 2.85e4·13-s + 4.85e4·14-s − 2.29e5·16-s + 4.47e5·17-s + 3.91e5·18-s − 5.28e5·19-s + 4.44e5·21-s − 1.53e5·22-s − 2.24e6·23-s − 2.36e6·24-s − 6.08e5·26-s − 2.54e5·27-s − 1.31e5·28-s + 5.98e6·29-s + 1.69e5·31-s + 1.32e6·32-s − 1.40e6·33-s + 9.54e6·34-s + ⋯ |
| L(s) = 1 | + 0.942·2-s + 1.39·3-s − 0.112·4-s + 1.31·6-s + 0.358·7-s − 1.04·8-s + 0.933·9-s − 0.147·11-s − 0.156·12-s − 0.277·13-s + 0.337·14-s − 0.874·16-s + 1.30·17-s + 0.879·18-s − 0.930·19-s + 0.498·21-s − 0.139·22-s − 1.67·23-s − 1.45·24-s − 0.261·26-s − 0.0921·27-s − 0.0403·28-s + 1.57·29-s + 0.0329·31-s + 0.223·32-s − 0.205·33-s + 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 21.3T + 512T^{2} \) |
| 3 | \( 1 - 195.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 2.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.17e3T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.69e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.76e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.18e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.14e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.40e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.88e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.92e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.14e9T + 7.60e17T^{2} \) |
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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
−9.545076880058830449598037768339, −8.376348336581657827835148668194, −8.080062502997130057589491677363, −6.62622685975630959623887848520, −5.45696926015440910854992501382, −4.43181291457768414397565722224, −3.55189754537267878883954406957, −2.75928007370150604573328908444, −1.68516957776384951383071061495, 0,
1.68516957776384951383071061495, 2.75928007370150604573328908444, 3.55189754537267878883954406957, 4.43181291457768414397565722224, 5.45696926015440910854992501382, 6.62622685975630959623887848520, 8.080062502997130057589491677363, 8.376348336581657827835148668194, 9.545076880058830449598037768339
