| L(s) = 1 | − 19.4·2-s − 193.·3-s − 133.·4-s − 1.26e3·5-s + 3.76e3·6-s − 1.84e3·7-s + 1.25e4·8-s + 1.77e4·9-s + 2.46e4·10-s − 4.03e4·11-s + 2.59e4·12-s − 8.80e4·13-s + 3.59e4·14-s + 2.45e5·15-s − 1.75e5·16-s − 3.45e5·18-s + 2.88e5·19-s + 1.69e5·20-s + 3.57e5·21-s + 7.84e5·22-s + 1.90e6·23-s − 2.43e6·24-s − 3.49e5·25-s + 1.71e6·26-s + 3.67e5·27-s + 2.47e5·28-s + 7.53e6·29-s + ⋯ |
| L(s) = 1 | − 0.859·2-s − 1.37·3-s − 0.261·4-s − 0.906·5-s + 1.18·6-s − 0.290·7-s + 1.08·8-s + 0.903·9-s + 0.778·10-s − 0.830·11-s + 0.360·12-s − 0.855·13-s + 0.249·14-s + 1.25·15-s − 0.670·16-s − 0.776·18-s + 0.507·19-s + 0.236·20-s + 0.401·21-s + 0.713·22-s + 1.41·23-s − 1.49·24-s − 0.178·25-s + 0.734·26-s + 0.133·27-s + 0.0759·28-s + 1.97·29-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
| \(L(5)\) |
\(\approx\) |
\(0.1085793165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1085793165\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 19.4T + 512T^{2} \) |
| 3 | \( 1 + 193.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.26e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.84e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.03e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.80e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 2.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.90e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.53e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.53e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.87e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.47e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.55e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.08e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.18e5T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.28e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.98e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.46e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.69e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.19e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.50e8T + 7.60e17T^{2} \) |
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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
−10.31306280746610609297397337410, −9.446674163547022940842985747094, −8.224627243267422745704348739928, −7.47017465682846092835034172363, −6.50071341054769915257808059207, −5.07016481983094483537665367402, −4.64570859181031048851207433686, −3.03874601151655973091120446233, −1.19329106913217343149694055413, −0.21176156859672893335134519412,
0.21176156859672893335134519412, 1.19329106913217343149694055413, 3.03874601151655973091120446233, 4.64570859181031048851207433686, 5.07016481983094483537665367402, 6.50071341054769915257808059207, 7.47017465682846092835034172363, 8.224627243267422745704348739928, 9.446674163547022940842985747094, 10.31306280746610609297397337410
