| L(s) = 1 | + 1.40·2-s − 3·3-s − 6.01·4-s + 5·5-s − 4.22·6-s + 22.4·7-s − 19.7·8-s + 9·9-s + 7.04·10-s + 11.9·11-s + 18.0·12-s − 24.3·13-s + 31.5·14-s − 15·15-s + 20.2·16-s − 57.0·17-s + 12.6·18-s − 101.·19-s − 30.0·20-s − 67.2·21-s + 16.8·22-s + 133.·23-s + 59.2·24-s + 25·25-s − 34.3·26-s − 27·27-s − 134.·28-s + ⋯ |
| L(s) = 1 | + 0.498·2-s − 0.577·3-s − 0.751·4-s + 0.447·5-s − 0.287·6-s + 1.21·7-s − 0.872·8-s + 0.333·9-s + 0.222·10-s + 0.327·11-s + 0.434·12-s − 0.520·13-s + 0.602·14-s − 0.258·15-s + 0.317·16-s − 0.814·17-s + 0.166·18-s − 1.22·19-s − 0.336·20-s − 0.698·21-s + 0.163·22-s + 1.20·23-s + 0.503·24-s + 0.200·25-s − 0.259·26-s − 0.192·27-s − 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.918550850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918550850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 1.40T + 8T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 - 11.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 133.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 490.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 320.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 612.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 569.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 689.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 125.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 356.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 947.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 597.T + 9.12e5T^{2} \) |
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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
−10.90105425819941109377575027626, −9.849260834025950715096617286944, −8.909804316935680525189515357544, −8.095981467397186707971108894432, −6.74243392330184941112889953837, −5.79396820707187029202082573448, −4.74269334162501638398106232719, −4.32976011783828265162166917631, −2.47698213947610411168312872873, −0.880889658107417387830054797693,
0.880889658107417387830054797693, 2.47698213947610411168312872873, 4.32976011783828265162166917631, 4.74269334162501638398106232719, 5.79396820707187029202082573448, 6.74243392330184941112889953837, 8.095981467397186707971108894432, 8.909804316935680525189515357544, 9.849260834025950715096617286944, 10.90105425819941109377575027626
