L(s) = 1 | − 3.69·2-s − 3·3-s + 5.68·4-s + 8.40·5-s + 11.0·6-s + 21.0·7-s + 8.55·8-s + 9·9-s − 31.0·10-s + 34.6·11-s − 17.0·12-s + 82.9·13-s − 77.9·14-s − 25.2·15-s − 77.1·16-s + 123.·17-s − 33.2·18-s + 135.·19-s + 47.8·20-s − 63.1·21-s − 128.·22-s + 42.0·23-s − 25.6·24-s − 54.3·25-s − 306.·26-s − 27·27-s + 119.·28-s + ⋯ |
L(s) = 1 | − 1.30·2-s − 0.577·3-s + 0.711·4-s + 0.751·5-s + 0.755·6-s + 1.13·7-s + 0.377·8-s + 0.333·9-s − 0.983·10-s + 0.950·11-s − 0.410·12-s + 1.76·13-s − 1.48·14-s − 0.433·15-s − 1.20·16-s + 1.76·17-s − 0.436·18-s + 1.63·19-s + 0.534·20-s − 0.656·21-s − 1.24·22-s + 0.381·23-s − 0.218·24-s − 0.435·25-s − 2.31·26-s − 0.192·27-s + 0.808·28-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
\(L(2)\) |
\(\approx\) |
\(1.388644389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388644389\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.69T + 8T^{2} \) |
| 5 | \( 1 - 8.40T + 125T^{2} \) |
| 7 | \( 1 - 21.0T + 343T^{2} \) |
| 11 | \( 1 - 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 42.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 331.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 15.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 218.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 307.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.90T + 3.00e5T^{2} \) |
| 71 | \( 1 + 184.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 945.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 787.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−10.42539484756719371080540583721, −9.706794158015346057128317405418, −8.900106217329100788094552489099, −8.032020378972260802321732356539, −7.18394528741657293141702881215, −5.96510036149672420102292995925, −5.15771148203817598074073703112, −3.67373805375715635814049443556, −1.45889398971130720020487726986, −1.18901924776635684537599767053,
1.18901924776635684537599767053, 1.45889398971130720020487726986, 3.67373805375715635814049443556, 5.15771148203817598074073703112, 5.96510036149672420102292995925, 7.18394528741657293141702881215, 8.032020378972260802321732356539, 8.900106217329100788094552489099, 9.706794158015346057128317405418, 10.42539484756719371080540583721