| L(s) = 1 | − 2·2-s − 8.60·3-s + 4·4-s + 0.977·5-s + 17.2·6-s + 26.8·7-s − 8·8-s + 47.0·9-s − 1.95·10-s + 37.8·11-s − 34.4·12-s + 67.8·13-s − 53.6·14-s − 8.41·15-s + 16·16-s + 26.7·17-s − 94.1·18-s + 52.8·19-s + 3.91·20-s − 230.·21-s − 75.6·22-s + 103.·23-s + 68.8·24-s − 124.·25-s − 135.·26-s − 172.·27-s + 107.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.65·3-s + 0.5·4-s + 0.0874·5-s + 1.17·6-s + 1.44·7-s − 0.353·8-s + 1.74·9-s − 0.0618·10-s + 1.03·11-s − 0.828·12-s + 1.44·13-s − 1.02·14-s − 0.144·15-s + 0.250·16-s + 0.381·17-s − 1.23·18-s + 0.638·19-s + 0.0437·20-s − 2.39·21-s − 0.732·22-s + 0.939·23-s + 0.585·24-s − 0.992·25-s − 1.02·26-s − 1.23·27-s + 0.724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.186021491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186021491\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 269 | \( 1 + 269T \) |
| good | 3 | \( 1 + 8.60T + 27T^{2} \) |
| 5 | \( 1 - 0.977T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 11 | \( 1 - 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 44.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 63.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 474.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 36.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 231.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 561.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 245.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−10.74154442230112592995237556322, −9.656429162741775003919539801626, −8.657844692267166008093178427562, −7.67572615483943247196251646406, −6.68683689596583351418894437764, −5.85216271616979485384299874929, −5.05006444359746864079907484517, −3.85641882587468177664289946305, −1.57380169961291688528678283148, −0.920809298679629953240507495640,
0.920809298679629953240507495640, 1.57380169961291688528678283148, 3.85641882587468177664289946305, 5.05006444359746864079907484517, 5.85216271616979485384299874929, 6.68683689596583351418894437764, 7.67572615483943247196251646406, 8.657844692267166008093178427562, 9.656429162741775003919539801626, 10.74154442230112592995237556322
