| L(s) = 1 | + 6.86·2-s − 207.·3-s − 464.·4-s + 1.87e3·5-s − 1.42e3·6-s − 3.27e3·7-s − 6.70e3·8-s + 2.35e4·9-s + 1.29e4·10-s − 6.06e4·11-s + 9.65e4·12-s − 7.41e4·13-s − 2.24e4·14-s − 3.90e5·15-s + 1.91e5·16-s + 1.61e5·18-s − 6.40e5·19-s − 8.73e5·20-s + 6.79e5·21-s − 4.16e5·22-s + 1.59e6·23-s + 1.39e6·24-s + 1.57e6·25-s − 5.09e5·26-s − 7.94e5·27-s + 1.52e6·28-s − 3.96e6·29-s + ⋯ |
| L(s) = 1 | + 0.303·2-s − 1.48·3-s − 0.907·4-s + 1.34·5-s − 0.449·6-s − 0.514·7-s − 0.579·8-s + 1.19·9-s + 0.408·10-s − 1.24·11-s + 1.34·12-s − 0.720·13-s − 0.156·14-s − 1.99·15-s + 0.732·16-s + 0.362·18-s − 1.12·19-s − 1.22·20-s + 0.762·21-s − 0.378·22-s + 1.18·23-s + 0.857·24-s + 0.807·25-s − 0.218·26-s − 0.287·27-s + 0.467·28-s − 1.04·29-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
| \(L(5)\) |
\(\approx\) |
\(0.1933043939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1933043939\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 6.86T + 512T^{2} \) |
| 3 | \( 1 + 207.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.87e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.06e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.41e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 6.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.59e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.96e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.74e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.52e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.43e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.35e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.35e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.55e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.12e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.57e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.63e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.53e9T + 7.60e17T^{2} \) |
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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
−10.15761649175881676503318648906, −9.680826834743209658232771762904, −8.474146581009871291816569548194, −6.88792499625254904071323201867, −6.04049113922778263864047559573, −5.21360423198110208039361470962, −4.82869210364704060433704762068, −3.10994220157460532074192378799, −1.71196560079911252697242743850, −0.19790462202944499522800158173,
0.19790462202944499522800158173, 1.71196560079911252697242743850, 3.10994220157460532074192378799, 4.82869210364704060433704762068, 5.21360423198110208039361470962, 6.04049113922778263864047559573, 6.88792499625254904071323201867, 8.474146581009871291816569548194, 9.680826834743209658232771762904, 10.15761649175881676503318648906
