| L(s) = 1 | + 80.9·2-s − 6.71e3·3-s − 1.24e5·4-s − 6.72e5·5-s − 5.43e5·6-s + 2.39e7·7-s − 2.06e7·8-s − 8.40e7·9-s − 5.44e7·10-s + 8.36e8·12-s + 4.52e9·13-s + 1.93e9·14-s + 4.51e9·15-s + 1.46e10·16-s + 8.38e9·17-s − 6.80e9·18-s + 4.14e10·19-s + 8.37e10·20-s − 1.60e11·21-s + 6.83e11·23-s + 1.38e11·24-s − 3.10e11·25-s + 3.65e11·26-s + 1.43e12·27-s − 2.98e12·28-s − 1.51e12·29-s + 3.65e11·30-s + ⋯ |
| L(s) = 1 | + 0.223·2-s − 0.590·3-s − 0.950·4-s − 0.769·5-s − 0.132·6-s + 1.57·7-s − 0.435·8-s − 0.650·9-s − 0.172·10-s + 0.561·12-s + 1.53·13-s + 0.351·14-s + 0.454·15-s + 0.852·16-s + 0.291·17-s − 0.145·18-s + 0.559·19-s + 0.731·20-s − 0.928·21-s + 1.82·23-s + 0.257·24-s − 0.407·25-s + 0.343·26-s + 0.975·27-s − 1.49·28-s − 0.561·29-s + 0.101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.662310203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662310203\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - 80.9T + 1.31e5T^{2} \) |
| 3 | \( 1 + 6.71e3T + 1.29e8T^{2} \) |
| 5 | \( 1 + 6.72e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 2.39e7T + 2.32e14T^{2} \) |
| 13 | \( 1 - 4.52e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 8.38e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 4.14e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.83e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.51e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.30e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.21e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.19e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.32e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.74e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.09e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.45e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.18e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 5.29e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.03e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 9.53e14T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.65e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.91e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.56e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 2.58e16T + 5.95e33T^{2} \) |
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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
−10.81617183910246299211489009588, −9.026709551329285350286592962898, −8.409874335413437096553470417951, −7.46366802689115426573712083239, −5.79704464489926155508090281488, −5.13605304252177149760604070745, −4.18407489730644015695941560804, −3.22331454552415514024299767752, −1.37860888962816517099678318727, −0.60290328238662148667196224744,
0.60290328238662148667196224744, 1.37860888962816517099678318727, 3.22331454552415514024299767752, 4.18407489730644015695941560804, 5.13605304252177149760604070745, 5.79704464489926155508090281488, 7.46366802689115426573712083239, 8.409874335413437096553470417951, 9.026709551329285350286592962898, 10.81617183910246299211489009588
