| L(s) = 1 | + 5.00·2-s − 3·3-s + 17.0·4-s + 16.1·5-s − 15.0·6-s − 23.8·7-s + 45.0·8-s + 9·9-s + 81.0·10-s + 9.48·11-s − 51.0·12-s + 45.8·13-s − 119.·14-s − 48.5·15-s + 89.4·16-s + 82.9·17-s + 45.0·18-s + 28.8·19-s + 275.·20-s + 71.4·21-s + 47.4·22-s + 104.·23-s − 135.·24-s + 137.·25-s + 229.·26-s − 27·27-s − 405.·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.577·3-s + 2.12·4-s + 1.44·5-s − 1.02·6-s − 1.28·7-s + 1.99·8-s + 0.333·9-s + 2.56·10-s + 0.259·11-s − 1.22·12-s + 0.977·13-s − 2.27·14-s − 0.836·15-s + 1.39·16-s + 1.18·17-s + 0.589·18-s + 0.348·19-s + 3.08·20-s + 0.742·21-s + 0.459·22-s + 0.949·23-s − 1.15·24-s + 1.09·25-s + 1.72·26-s − 0.192·27-s − 2.73·28-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
| \(L(2)\) |
\(\approx\) |
\(5.855964428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.855964428\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.00T + 8T^{2} \) |
| 5 | \( 1 - 16.1T + 125T^{2} \) |
| 7 | \( 1 + 23.8T + 343T^{2} \) |
| 11 | \( 1 - 9.48T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 35.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 494.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 244.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 173.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 172.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 899.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 698.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 295.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 659.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 816.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 71.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 51.1T + 9.12e5T^{2} \) |
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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
−10.75695669070917665110347203784, −10.01325106522824941741958596096, −9.086464060808543293661905936664, −7.15595192843870140620640462028, −6.35794010671049439526363770657, −5.82553580896541577195694774214, −5.14991079013964398196311003747, −3.72174938953010187556593716811, −2.87743461466229070306438060034, −1.41732384710947766606580971966,
1.41732384710947766606580971966, 2.87743461466229070306438060034, 3.72174938953010187556593716811, 5.14991079013964398196311003747, 5.82553580896541577195694774214, 6.35794010671049439526363770657, 7.15595192843870140620640462028, 9.086464060808543293661905936664, 10.01325106522824941741958596096, 10.75695669070917665110347203784
