| L(s) = 1 | − 8·2-s − 75.8·3-s + 64·4-s + 606.·6-s + 343·7-s − 512·8-s + 3.56e3·9-s − 8.25e3·11-s − 4.85e3·12-s + 1.09e4·13-s − 2.74e3·14-s + 4.09e3·16-s + 5.41e3·17-s − 2.85e4·18-s + 1.65e4·19-s − 2.60e4·21-s + 6.60e4·22-s + 1.06e5·23-s + 3.88e4·24-s − 8.76e4·26-s − 1.04e5·27-s + 2.19e4·28-s − 2.13e5·29-s + 2.29e5·31-s − 3.27e4·32-s + 6.26e5·33-s − 4.33e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·3-s + 0.5·4-s + 1.14·6-s + 0.377·7-s − 0.353·8-s + 1.63·9-s − 1.87·11-s − 0.810·12-s + 1.38·13-s − 0.267·14-s + 0.250·16-s + 0.267·17-s − 1.15·18-s + 0.554·19-s − 0.612·21-s + 1.32·22-s + 1.82·23-s + 0.573·24-s − 0.977·26-s − 1.02·27-s + 0.188·28-s − 1.62·29-s + 1.38·31-s − 0.176·32-s + 3.03·33-s − 0.189·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7136243117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7136243117\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 + 75.8T + 2.18e3T^{2} \) |
| 11 | \( 1 + 8.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.09e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 5.41e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.13e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.53e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.76e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.17e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.54e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.95e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.50e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.50e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.27e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.82e6T + 8.07e13T^{2} \) |
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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
−10.56778489825384519224523622481, −9.611829927656063923452304685510, −8.287089155343474314916574774025, −7.47497978491723015535827435865, −6.37984616040213395165950427682, −5.52501392623993314256563471894, −4.79187026826847886812162296606, −3.04432947260543557514372916221, −1.41804992161918704592283297208, −0.52802687522186568823891871334,
0.52802687522186568823891871334, 1.41804992161918704592283297208, 3.04432947260543557514372916221, 4.79187026826847886812162296606, 5.52501392623993314256563471894, 6.37984616040213395165950427682, 7.47497978491723015535827435865, 8.287089155343474314916574774025, 9.611829927656063923452304685510, 10.56778489825384519224523622481
