| L(s) = 1 | − 354.·2-s − 6.56e3·3-s − 5.26e3·4-s + 2.32e6·6-s − 1.41e7·7-s + 4.83e7·8-s + 4.30e7·9-s + 5.09e8·11-s + 3.45e7·12-s + 4.37e9·13-s + 5.01e9·14-s − 1.64e10·16-s + 1.49e10·17-s − 1.52e10·18-s + 1.24e11·19-s + 9.28e10·21-s − 1.80e11·22-s + 8.35e10·23-s − 3.17e11·24-s − 1.55e12·26-s − 2.82e11·27-s + 7.44e10·28-s + 5.15e12·29-s + 3.28e12·31-s − 4.99e11·32-s − 3.34e12·33-s − 5.28e12·34-s + ⋯ |
| L(s) = 1 | − 0.979·2-s − 0.577·3-s − 0.0401·4-s + 0.565·6-s − 0.927·7-s + 1.01·8-s + 0.333·9-s + 0.717·11-s + 0.0231·12-s + 1.48·13-s + 0.908·14-s − 0.958·16-s + 0.518·17-s − 0.326·18-s + 1.68·19-s + 0.535·21-s − 0.702·22-s + 0.222·23-s − 0.588·24-s − 1.45·26-s − 0.192·27-s + 0.0372·28-s + 1.91·29-s + 0.691·31-s − 0.0802·32-s − 0.414·33-s − 0.507·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.234878730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.234878730\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 354.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 1.41e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 5.09e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.37e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.49e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.24e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 8.35e10T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.15e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.28e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.60e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 4.59e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.93e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.62e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.27e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 7.24e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.08e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.17e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.82e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 4.08e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.19e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.80e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.90e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.01e17T + 5.95e33T^{2} \) |
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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
−10.97487274732219114074502969453, −9.931669619059458488564411901248, −9.215038719853697950937014479219, −8.073246908819543754458832817903, −6.83473428636481478544791597101, −5.84182241161276742512882414524, −4.37555768114059551669938346923, −3.16185716201813797225808814695, −1.17904517152618287297441515767, −0.77413795803851714676572968180,
0.77413795803851714676572968180, 1.17904517152618287297441515767, 3.16185716201813797225808814695, 4.37555768114059551669938346923, 5.84182241161276742512882414524, 6.83473428636481478544791597101, 8.073246908819543754458832817903, 9.215038719853697950937014479219, 9.931669619059458488564411901248, 10.97487274732219114074502969453
