| L(s) = 1 | − 3.62·2-s − 3·3-s + 5.11·4-s − 9.00·5-s + 10.8·6-s + 5.02·7-s + 10.4·8-s + 9·9-s + 32.6·10-s + 49.7·11-s − 15.3·12-s + 36.3·13-s − 18.1·14-s + 27.0·15-s − 78.7·16-s − 71.5·17-s − 32.5·18-s − 154.·19-s − 46.0·20-s − 15.0·21-s − 180.·22-s + 120.·23-s − 31.3·24-s − 43.8·25-s − 131.·26-s − 27·27-s + 25.6·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.638·4-s − 0.805·5-s + 0.739·6-s + 0.271·7-s + 0.462·8-s + 0.333·9-s + 1.03·10-s + 1.36·11-s − 0.368·12-s + 0.775·13-s − 0.347·14-s + 0.465·15-s − 1.23·16-s − 1.02·17-s − 0.426·18-s − 1.86·19-s − 0.514·20-s − 0.156·21-s − 1.74·22-s + 1.09·23-s − 0.266·24-s − 0.350·25-s − 0.993·26-s − 0.192·27-s + 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5658995600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5658995600\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 + 3.62T + 8T^{2} \) |
| 5 | \( 1 + 9.00T + 125T^{2} \) |
| 7 | \( 1 - 5.02T + 343T^{2} \) |
| 11 | \( 1 - 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 15.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 5.10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 217.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 550.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 736.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 644.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 12.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−10.78597921046106449792819565418, −9.594905220170104030598241675676, −8.682229494915391197113481973784, −8.238193072675422516814257647493, −6.93323975656313763123685198089, −6.42357601239973772927094676641, −4.67361766035267752388111946363, −3.90691750393716367112615550665, −1.81154857197273571839835220288, −0.60657107491118381351711722193,
0.60657107491118381351711722193, 1.81154857197273571839835220288, 3.90691750393716367112615550665, 4.67361766035267752388111946363, 6.42357601239973772927094676641, 6.93323975656313763123685198089, 8.238193072675422516814257647493, 8.682229494915391197113481973784, 9.594905220170104030598241675676, 10.78597921046106449792819565418
