| L(s) = 1 | − 10·2-s − 28·4-s − 472.·5-s + 1.56e3·8-s + 4.72e3·10-s + 4.74e3·11-s + 3.30e3·13-s − 1.20e4·16-s + 3.49e4·17-s − 2.60e4·19-s + 1.32e4·20-s − 4.74e4·22-s + 7.55e4·23-s + 1.45e5·25-s − 3.30e4·26-s + 1.09e5·29-s + 4.72e3·31-s − 7.95e4·32-s − 3.49e5·34-s + 1.99e5·37-s + 2.60e5·38-s − 7.37e5·40-s − 4.96e5·41-s − 4.19e5·43-s − 1.32e5·44-s − 7.55e5·46-s − 2.22e5·47-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 0.218·4-s − 1.69·5-s + 1.07·8-s + 1.49·10-s + 1.07·11-s + 0.417·13-s − 0.733·16-s + 1.72·17-s − 0.869·19-s + 0.370·20-s − 0.950·22-s + 1.29·23-s + 1.86·25-s − 0.369·26-s + 0.832·29-s + 0.0285·31-s − 0.428·32-s − 1.52·34-s + 0.647·37-s + 0.768·38-s − 1.82·40-s − 1.12·41-s − 0.804·43-s − 0.235·44-s − 1.14·46-s − 0.312·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9177977543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9177977543\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 10T + 128T^{2} \) |
| 5 | \( 1 + 472.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.74e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.30e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.60e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.55e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.72e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.99e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.22e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.09e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.72e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.94e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.98e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.83e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7T + 8.07e13T^{2} \) |
| 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
−9.879682691618393744087486153928, −8.808526932953341907225274315962, −8.289604911112917044900830669436, −7.49611964227834324105209248786, −6.62770124298347243825807356265, −4.99109932450502722430793876187, −4.05485969267863083728906692259, −3.29053054461208711614918842444, −1.29792232058205845365416560198, −0.58504588694935449130304982919,
0.58504588694935449130304982919, 1.29792232058205845365416560198, 3.29053054461208711614918842444, 4.05485969267863083728906692259, 4.99109932450502722430793876187, 6.62770124298347243825807356265, 7.49611964227834324105209248786, 8.289604911112917044900830669436, 8.808526932953341907225274315962, 9.879682691618393744087486153928
