| L(s) = 1 | + 16·2-s − 47.7·3-s + 256·4-s − 625·5-s − 764.·6-s − 2.40e3·7-s + 4.09e3·8-s − 1.73e4·9-s − 1.00e4·10-s + 4.67e4·11-s − 1.22e4·12-s + 3.02e4·13-s − 3.84e4·14-s + 2.98e4·15-s + 6.55e4·16-s + 2.96e5·17-s − 2.78e5·18-s + 1.84e5·19-s − 1.60e5·20-s + 1.14e5·21-s + 7.48e5·22-s − 2.11e4·23-s − 1.95e5·24-s + 3.90e5·25-s + 4.83e5·26-s + 1.77e6·27-s − 6.14e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.340·3-s + 0.5·4-s − 0.447·5-s − 0.240·6-s − 0.377·7-s + 0.353·8-s − 0.883·9-s − 0.316·10-s + 0.963·11-s − 0.170·12-s + 0.293·13-s − 0.267·14-s + 0.152·15-s + 0.250·16-s + 0.860·17-s − 0.625·18-s + 0.324·19-s − 0.223·20-s + 0.128·21-s + 0.681·22-s − 0.0157·23-s − 0.120·24-s + 0.200·25-s + 0.207·26-s + 0.641·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.514912860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.514912860\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 + 47.7T + 1.96e4T^{2} \) |
| 11 | \( 1 - 4.67e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.02e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.84e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.11e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.53e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.41e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.47e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.09e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.08e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.45e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.87e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.97e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.13e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.41e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.98e8T + 7.60e17T^{2} \) |
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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
−12.66868329805969914778264325493, −11.79606022405377849289258439213, −10.97749621547641807833159831023, −9.457729365444238835209112547645, −8.021823315338662370129275237130, −6.59342515204399879626648990393, −5.59219077051837557249380421684, −4.09933221612695671803949746649, −2.89488284113051692185814811104, −0.900313268931347071650415188452,
0.900313268931347071650415188452, 2.89488284113051692185814811104, 4.09933221612695671803949746649, 5.59219077051837557249380421684, 6.59342515204399879626648990393, 8.021823315338662370129275237130, 9.457729365444238835209112547645, 10.97749621547641807833159831023, 11.79606022405377849289258439213, 12.66868329805969914778264325493
