| L(s) = 1 | + 4.24·2-s − 3·3-s + 9.99·4-s + 19.9·5-s − 12.7·6-s − 20.9·7-s + 8.48·8-s + 9·9-s + 84.7·10-s + 16.0·11-s − 29.9·12-s − 34.9·13-s − 88.9·14-s − 59.9·15-s − 44.0·16-s − 17·17-s + 38.1·18-s − 80.8·19-s + 199.·20-s + 62.9·21-s + 68.0·22-s − 115.·23-s − 25.4·24-s + 273.·25-s − 148.·26-s − 27·27-s − 209.·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.577·3-s + 1.24·4-s + 1.78·5-s − 0.866·6-s − 1.13·7-s + 0.374·8-s + 0.333·9-s + 2.67·10-s + 0.439·11-s − 0.721·12-s − 0.745·13-s − 1.69·14-s − 1.03·15-s − 0.687·16-s − 0.242·17-s + 0.500·18-s − 0.976·19-s + 2.23·20-s + 0.653·21-s + 0.659·22-s − 1.05·23-s − 0.216·24-s + 2.19·25-s − 1.11·26-s − 0.192·27-s − 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.636758209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.636758209\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 - 4.24T + 8T^{2} \) |
| 5 | \( 1 - 19.9T + 125T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 - 16.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 629.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 461.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 686.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 254T + 4.93e5T^{2} \) |
| 83 | \( 1 - 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 732.T + 9.12e5T^{2} \) |
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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
−14.62362983344960776055261691856, −13.68222288785837112621247610811, −12.93585579326445167328565546576, −12.06647203562279580501175738999, −10.35686539836264492321707011189, −9.408194808190971945735719944340, −6.42113878209371864504508109495, −6.11885099356846956181751603264, −4.61604693073622875217041840227, −2.55702699647835084381266876097,
2.55702699647835084381266876097, 4.61604693073622875217041840227, 6.11885099356846956181751603264, 6.42113878209371864504508109495, 9.408194808190971945735719944340, 10.35686539836264492321707011189, 12.06647203562279580501175738999, 12.93585579326445167328565546576, 13.68222288785837112621247610811, 14.62362983344960776055261691856
