| L(s) = 1 | + 12.9·2-s + 101.·3-s − 345.·4-s + 2.07e3·5-s + 1.30e3·6-s + 2.16e3·7-s − 1.10e4·8-s − 9.39e3·9-s + 2.68e4·10-s − 1.31e4·11-s − 3.50e4·12-s − 1.78e4·13-s + 2.79e4·14-s + 2.10e5·15-s + 3.37e4·16-s − 1.21e5·18-s − 3.09e4·19-s − 7.17e5·20-s + 2.19e5·21-s − 1.69e5·22-s − 1.93e5·23-s − 1.12e6·24-s + 2.36e6·25-s − 2.30e5·26-s − 2.94e6·27-s − 7.46e5·28-s + 1.83e6·29-s + ⋯ |
| L(s) = 1 | + 0.570·2-s + 0.722·3-s − 0.674·4-s + 1.48·5-s + 0.412·6-s + 0.340·7-s − 0.955·8-s − 0.477·9-s + 0.848·10-s − 0.270·11-s − 0.487·12-s − 0.173·13-s + 0.194·14-s + 1.07·15-s + 0.128·16-s − 0.272·18-s − 0.0544·19-s − 1.00·20-s + 0.245·21-s − 0.154·22-s − 0.144·23-s − 0.690·24-s + 1.21·25-s − 0.0989·26-s − 1.06·27-s − 0.229·28-s + 0.481·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 12.9T + 512T^{2} \) |
| 3 | \( 1 - 101.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.07e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.78e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.09e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.93e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.83e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.25e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.78e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.91e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.24e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.01e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.14e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 9.58e6T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.70e8T + 7.60e17T^{2} \) |
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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
−9.634195939894776432241738481930, −8.867754389255386954216636147219, −8.137591763232360392624197275825, −6.58037394115002522803927228016, −5.55733382845708236919080087195, −4.93990138788358096997249556754, −3.53075423318275244081837875798, −2.58413261953093758280047770067, −1.59257447430119190263179051554, 0,
1.59257447430119190263179051554, 2.58413261953093758280047770067, 3.53075423318275244081837875798, 4.93990138788358096997249556754, 5.55733382845708236919080087195, 6.58037394115002522803927228016, 8.137591763232360392624197275825, 8.867754389255386954216636147219, 9.634195939894776432241738481930
