| L(s) = 1 | − 3.15·2-s + 136.·3-s − 502.·4-s − 430.·6-s − 9.39e3·7-s + 3.19e3·8-s − 1.04e3·9-s + 4.40e4·11-s − 6.85e4·12-s − 2.85e4·13-s + 2.96e4·14-s + 2.46e5·16-s − 2.82e4·17-s + 3.28e3·18-s + 2.73e5·19-s − 1.28e6·21-s − 1.38e5·22-s + 1.12e6·23-s + 4.36e5·24-s + 8.99e4·26-s − 2.82e6·27-s + 4.71e6·28-s − 1.63e6·29-s + 6.65e6·31-s − 2.41e6·32-s + 6.02e6·33-s + 8.91e4·34-s + ⋯ |
| L(s) = 1 | − 0.139·2-s + 0.973·3-s − 0.980·4-s − 0.135·6-s − 1.47·7-s + 0.275·8-s − 0.0529·9-s + 0.908·11-s − 0.954·12-s − 0.277·13-s + 0.206·14-s + 0.942·16-s − 0.0821·17-s + 0.00736·18-s + 0.482·19-s − 1.44·21-s − 0.126·22-s + 0.841·23-s + 0.268·24-s + 0.0386·26-s − 1.02·27-s + 1.45·28-s − 0.429·29-s + 1.29·31-s − 0.406·32-s + 0.883·33-s + 0.0114·34-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 | \( 1 + 3.15T + 512T^{2} \) |
| 3 | \( 1 - 136.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 9.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.40e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.82e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.63e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.15e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.48e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.00e9T + 7.60e17T^{2} \) |
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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
−9.479032626511805619927500038555, −8.873624881712974556897249705902, −7.940541951584335310110746159534, −6.80547930032988573506980598949, −5.74984256741546881694190556876, −4.37158216039019671758346243627, −3.44761531244523069042544151461, −2.72460030801744211950827356308, −1.09082487369083514972935820022, 0,
1.09082487369083514972935820022, 2.72460030801744211950827356308, 3.44761531244523069042544151461, 4.37158216039019671758346243627, 5.74984256741546881694190556876, 6.80547930032988573506980598949, 7.940541951584335310110746159534, 8.873624881712974556897249705902, 9.479032626511805619927500038555
