| L(s) = 1 | − 19.6·3-s − 46.7·5-s − 49·7-s + 143.·9-s − 666.·11-s − 650.·13-s + 918.·15-s + 1.18e3·17-s + 1.56e3·19-s + 962.·21-s + 1.10e3·23-s − 939.·25-s + 1.96e3·27-s + 2.39e3·29-s + 2.04e3·31-s + 1.30e4·33-s + 2.29e3·35-s + 1.07e3·37-s + 1.27e4·39-s + 1.09e3·41-s − 1.65e4·43-s − 6.68e3·45-s + 8.29e3·47-s + 2.40e3·49-s − 2.33e4·51-s + 5.51e3·53-s + 3.11e4·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.836·5-s − 0.377·7-s + 0.588·9-s − 1.65·11-s − 1.06·13-s + 1.05·15-s + 0.996·17-s + 0.994·19-s + 0.476·21-s + 0.433·23-s − 0.300·25-s + 0.518·27-s + 0.529·29-s + 0.382·31-s + 2.09·33-s + 0.316·35-s + 0.129·37-s + 1.34·39-s + 0.102·41-s − 1.36·43-s − 0.492·45-s + 0.547·47-s + 0.142·49-s − 1.25·51-s + 0.269·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4925006273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4925006273\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 19.6T + 243T^{2} \) |
| 5 | \( 1 + 46.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 650.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.51e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 675.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.29e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.30208050176208976936281176097, −11.81357685016776800200109602500, −10.65909771382784445835039782122, −9.852044866254439594247604416569, −8.027405614223930089162366475093, −7.15426099659702670297806941846, −5.64810768471317763730689485251, −4.83452450003376614175435518495, −3.00532177992967517781762274617, −0.50091096055834706358739441815,
0.50091096055834706358739441815, 3.00532177992967517781762274617, 4.83452450003376614175435518495, 5.64810768471317763730689485251, 7.15426099659702670297806941846, 8.027405614223930089162366475093, 9.852044866254439594247604416569, 10.65909771382784445835039782122, 11.81357685016776800200109602500, 12.30208050176208976936281176097
