| L(s) = 1 | + 21.4·2-s − 57.9·3-s + 332.·4-s + 368.·5-s − 1.24e3·6-s + 343·7-s + 4.38e3·8-s + 1.17e3·9-s + 7.89e3·10-s − 3.95e3·11-s − 1.92e4·12-s + 2.19e3·13-s + 7.35e3·14-s − 2.13e4·15-s + 5.15e4·16-s + 1.84e4·17-s + 2.51e4·18-s + 4.95e4·19-s + 1.22e5·20-s − 1.98e4·21-s − 8.48e4·22-s − 6.58e4·23-s − 2.54e5·24-s + 5.73e4·25-s + 4.71e4·26-s + 5.88e4·27-s + 1.13e5·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 1.23·3-s + 2.59·4-s + 1.31·5-s − 2.35·6-s + 0.377·7-s + 3.02·8-s + 0.535·9-s + 2.49·10-s − 0.896·11-s − 3.21·12-s + 0.277·13-s + 0.716·14-s − 1.63·15-s + 3.14·16-s + 0.908·17-s + 1.01·18-s + 1.65·19-s + 3.41·20-s − 0.468·21-s − 1.69·22-s − 1.12·23-s − 3.75·24-s + 0.733·25-s + 0.525·26-s + 0.575·27-s + 0.981·28-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
| \(L(4)\) |
\(\approx\) |
\(5.749071594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.749071594\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 21.4T + 128T^{2} \) |
| 3 | \( 1 + 57.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 368.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.95e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.95e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.39e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 8.05e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.00e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.94e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.72e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.55e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.73e6T + 8.07e13T^{2} \) |
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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
−12.65449290265815992181776509344, −11.93195699344654610313192907639, −10.91219063446376968516331136963, −10.05691976420307511409570984567, −7.49509251328973225335252742969, −6.04038483809205266684210711014, −5.64245754513886897928545409723, −4.78234011268111345776038922285, −2.94674567585944454400392899466, −1.46985071615891315727496169394,
1.46985071615891315727496169394, 2.94674567585944454400392899466, 4.78234011268111345776038922285, 5.64245754513886897928545409723, 6.04038483809205266684210711014, 7.49509251328973225335252742969, 10.05691976420307511409570984567, 10.91219063446376968516331136963, 11.93195699344654610313192907639, 12.65449290265815992181776509344
