| L(s) = 1 | + 8·2-s − 89.8·3-s + 64·4-s + 433.·5-s − 718.·6-s − 1.00e3·7-s + 512·8-s + 5.87e3·9-s + 3.47e3·10-s − 5.74e3·12-s + 2.78e3·13-s − 8.00e3·14-s − 3.89e4·15-s + 4.09e3·16-s + 2.62e4·17-s + 4.70e4·18-s − 5.60e3·19-s + 2.77e4·20-s + 8.98e4·21-s − 3.77e4·23-s − 4.59e4·24-s + 1.10e5·25-s + 2.22e4·26-s − 3.31e5·27-s − 6.40e4·28-s − 1.52e5·29-s − 3.11e5·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.92·3-s + 0.5·4-s + 1.55·5-s − 1.35·6-s − 1.10·7-s + 0.353·8-s + 2.68·9-s + 1.09·10-s − 0.960·12-s + 0.351·13-s − 0.779·14-s − 2.98·15-s + 0.250·16-s + 1.29·17-s + 1.90·18-s − 0.187·19-s + 0.776·20-s + 2.11·21-s − 0.646·23-s − 0.678·24-s + 1.40·25-s + 0.248·26-s − 3.24·27-s − 0.551·28-s − 1.15·29-s − 2.10·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.159452644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159452644\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 89.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 433.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.00e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 2.78e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.62e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.60e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.42e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.57e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.28e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.63e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.71e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.91e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.75e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.03e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.36e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.46e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.19e7T + 8.07e13T^{2} \) |
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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
−10.85862242944168363605763522152, −10.12368632630507571444730911653, −9.519949479978266945511576877662, −7.25561589711709771407245587284, −6.24540515675881529757967627615, −5.87330110980417967851337694393, −5.13815126456607991716183920408, −3.69443292907047669416552613139, −1.94876795420769315957560457050, −0.75842019811135952549877905607,
0.75842019811135952549877905607, 1.94876795420769315957560457050, 3.69443292907047669416552613139, 5.13815126456607991716183920408, 5.87330110980417967851337694393, 6.24540515675881529757967627615, 7.25561589711709771407245587284, 9.519949479978266945511576877662, 10.12368632630507571444730911653, 10.85862242944168363605763522152
