| L(s) = 1 | + 1.26·2-s + 7.25·3-s − 6.41·4-s − 18.6·5-s + 9.14·6-s − 13.1·7-s − 18.1·8-s + 25.6·9-s − 23.5·10-s − 7.55·11-s − 46.5·12-s + 81.2·13-s − 16.5·14-s − 135.·15-s + 28.4·16-s + 49.6·17-s + 32.2·18-s + 107.·19-s + 119.·20-s − 95.4·21-s − 9.52·22-s − 131.·24-s + 224.·25-s + 102.·26-s − 10.0·27-s + 84.3·28-s + 169.·29-s + ⋯ |
| L(s) = 1 | + 0.445·2-s + 1.39·3-s − 0.801·4-s − 1.67·5-s + 0.622·6-s − 0.710·7-s − 0.802·8-s + 0.948·9-s − 0.745·10-s − 0.207·11-s − 1.11·12-s + 1.73·13-s − 0.316·14-s − 2.33·15-s + 0.443·16-s + 0.708·17-s + 0.422·18-s + 1.30·19-s + 1.34·20-s − 0.991·21-s − 0.0923·22-s − 1.12·24-s + 1.79·25-s + 0.772·26-s − 0.0713·27-s + 0.569·28-s + 1.08·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.178420214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178420214\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 1.26T + 8T^{2} \) |
| 3 | \( 1 - 7.25T + 27T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 7 | \( 1 + 13.1T + 343T^{2} \) |
| 11 | \( 1 + 7.55T + 1.33e3T^{2} \) |
| 13 | \( 1 - 81.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.22T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 518.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 566.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 278.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 776.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 68.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 144.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 828.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−10.28004947581005372295201791106, −9.254508667212735316263513129229, −8.520903712615492899735242308045, −8.054145554905381128449327250236, −7.07860852991847288554128890367, −5.63955279565750497064912806963, −4.26084334264517794294060284793, −3.45142614461036190258448535210, −3.19180851772849460524609926150, −0.810088659832649658577615559583,
0.810088659832649658577615559583, 3.19180851772849460524609926150, 3.45142614461036190258448535210, 4.26084334264517794294060284793, 5.63955279565750497064912806963, 7.07860852991847288554128890367, 8.054145554905381128449327250236, 8.520903712615492899735242308045, 9.254508667212735316263513129229, 10.28004947581005372295201791106
