| L(s) = 1 | + 16.6·2-s − 6.69·3-s + 150.·4-s + 406.·5-s − 111.·6-s − 315.·7-s + 370.·8-s − 2.14e3·9-s + 6.78e3·10-s − 491.·11-s − 1.00e3·12-s − 2.19e3·13-s − 5.25e3·14-s − 2.72e3·15-s − 1.30e4·16-s − 1.77e4·17-s − 3.57e4·18-s + 3.23e4·19-s + 6.11e4·20-s + 2.11e3·21-s − 8.20e3·22-s + 7.06e4·23-s − 2.48e3·24-s + 8.73e4·25-s − 3.66e4·26-s + 2.89e4·27-s − 4.73e4·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.143·3-s + 1.17·4-s + 1.45·5-s − 0.210·6-s − 0.347·7-s + 0.256·8-s − 0.979·9-s + 2.14·10-s − 0.111·11-s − 0.167·12-s − 0.277·13-s − 0.512·14-s − 0.208·15-s − 0.796·16-s − 0.877·17-s − 1.44·18-s + 1.08·19-s + 1.70·20-s + 0.0497·21-s − 0.164·22-s + 1.21·23-s − 0.0366·24-s + 1.11·25-s − 0.408·26-s + 0.283·27-s − 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.017273810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.017273810\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 - 16.6T + 128T^{2} \) |
| 3 | \( 1 + 6.69T + 2.18e3T^{2} \) |
| 5 | \( 1 - 406.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 315.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 491.T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.77e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.92e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.93e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.95e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.93e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.26e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.77e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.74e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.96e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.85e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.92e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.72e6T + 8.07e13T^{2} \) |
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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
−18.05679665927669408186667886240, −16.80026399135479113004494625496, −15.01871169722274080196698145152, −13.81999604744391811100240295971, −13.04623553600852369336762826649, −11.38085626458845897575874070244, −9.410045318746754653040104693118, −6.35222301081581603146571835495, −5.20461931369148176117106558577, −2.73096935418242692026590505055,
2.73096935418242692026590505055, 5.20461931369148176117106558577, 6.35222301081581603146571835495, 9.410045318746754653040104693118, 11.38085626458845897575874070244, 13.04623553600852369336762826649, 13.81999604744391811100240295971, 15.01871169722274080196698145152, 16.80026399135479113004494625496, 18.05679665927669408186667886240
