| L(s) = 1 | + 5.06·2-s + 91.5·3-s − 102.·4-s − 15.8·5-s + 463.·6-s + 343·7-s − 1.16e3·8-s + 6.19e3·9-s − 80.5·10-s − 579.·11-s − 9.36e3·12-s + 2.19e3·13-s + 1.73e3·14-s − 1.45e3·15-s + 7.17e3·16-s + 2.59e4·17-s + 3.13e4·18-s + 5.67e4·19-s + 1.62e3·20-s + 3.13e4·21-s − 2.93e3·22-s + 1.71e4·23-s − 1.06e5·24-s − 7.78e4·25-s + 1.11e4·26-s + 3.66e5·27-s − 3.50e4·28-s + ⋯ |
| L(s) = 1 | + 0.448·2-s + 1.95·3-s − 0.799·4-s − 0.0568·5-s + 0.876·6-s + 0.377·7-s − 0.806·8-s + 2.83·9-s − 0.0254·10-s − 0.131·11-s − 1.56·12-s + 0.277·13-s + 0.169·14-s − 0.111·15-s + 0.437·16-s + 1.28·17-s + 1.26·18-s + 1.89·19-s + 0.0454·20-s + 0.739·21-s − 0.0587·22-s + 0.293·23-s − 1.57·24-s − 0.996·25-s + 0.124·26-s + 3.58·27-s − 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.472242782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.472242782\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 5.06T + 128T^{2} \) |
| 3 | \( 1 - 91.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 15.8T + 7.81e4T^{2} \) |
| 11 | \( 1 + 579.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.67e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.71e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.47e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.56e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.86e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.99e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.26e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.81e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.77e6T + 8.07e13T^{2} \) |
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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
−13.21617165068745411664291797352, −11.98468323624242149189932443756, −9.958574289516056350248779225241, −9.312040463911508798050286439494, −8.219394093027801447613748356538, −7.46933215813166924124029306664, −5.27495524703005699123453113375, −3.84969856016893263449107447326, −3.08258049307173903683321344673, −1.36877779662944549026067084814,
1.36877779662944549026067084814, 3.08258049307173903683321344673, 3.84969856016893263449107447326, 5.27495524703005699123453113375, 7.46933215813166924124029306664, 8.219394093027801447613748356538, 9.312040463911508798050286439494, 9.958574289516056350248779225241, 11.98468323624242149189932443756, 13.21617165068745411664291797352
