| L(s) = 1 | − 24.7·2-s + 194.·3-s + 99.0·4-s − 920.·5-s − 4.80e3·6-s + 5.35e3·7-s + 1.02e4·8-s + 1.80e4·9-s + 2.27e4·10-s + 7.92e4·11-s + 1.92e4·12-s + 2.85e4·13-s − 1.32e5·14-s − 1.78e5·15-s − 3.03e5·16-s + 4.52e5·17-s − 4.46e5·18-s + 2.12e5·19-s − 9.11e4·20-s + 1.04e6·21-s − 1.95e6·22-s − 7.59e5·23-s + 1.98e6·24-s − 1.10e6·25-s − 7.05e5·26-s − 3.15e5·27-s + 5.30e5·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 1.38·3-s + 0.193·4-s − 0.658·5-s − 1.51·6-s + 0.843·7-s + 0.881·8-s + 0.917·9-s + 0.719·10-s + 1.63·11-s + 0.267·12-s + 0.277·13-s − 0.921·14-s − 0.911·15-s − 1.15·16-s + 1.31·17-s − 1.00·18-s + 0.374·19-s − 0.127·20-s + 1.16·21-s − 1.78·22-s − 0.565·23-s + 1.22·24-s − 0.566·25-s − 0.302·26-s − 0.114·27-s + 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.426438834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426438834\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 24.7T + 512T^{2} \) |
| 3 | \( 1 - 194.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 920.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.92e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.52e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.59e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 9.00e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.39e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.33e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.14e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.15e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.44e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.03e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.20e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.91e8T + 7.60e17T^{2} \) |
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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
−17.92982341369144045536226323469, −16.51473936603934148706872001037, −14.78732209468622680786321621139, −13.90125936530475004012904538721, −11.65231741353258348395217705382, −9.689771883234677143779068138526, −8.521660255554272723527879648770, −7.63330555332640379495250506575, −3.89953980598280880035062853658, −1.41442314638227447995018807528,
1.41442314638227447995018807528, 3.89953980598280880035062853658, 7.63330555332640379495250506575, 8.521660255554272723527879648770, 9.689771883234677143779068138526, 11.65231741353258348395217705382, 13.90125936530475004012904538721, 14.78732209468622680786321621139, 16.51473936603934148706872001037, 17.92982341369144045536226323469
