| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5.17·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 10.3·10-s + 14.7·11-s + 12·12-s − 13·13-s + 14·14-s − 15.5·15-s + 16·16-s − 70.5·17-s − 18·18-s + 128.·19-s − 20.7·20-s − 21·21-s − 29.4·22-s + 83.1·23-s − 24·24-s − 98.1·25-s + 26·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.463·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.327·10-s + 0.403·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.267·15-s + 0.250·16-s − 1.00·17-s − 0.235·18-s + 1.54·19-s − 0.231·20-s − 0.218·21-s − 0.285·22-s + 0.753·23-s − 0.204·24-s − 0.785·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 5.17T + 125T^{2} \) |
| 11 | \( 1 - 14.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 70.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 236.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 314.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 375.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 416.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 60.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 448.T + 9.12e5T^{2} \) |
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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
−9.605284750774727919276580183024, −9.231213262047884386489035705174, −8.206422999833152166964222667455, −7.37289794031841749582743307472, −6.66814066786533764008775580892, −5.30256366122942019045428068996, −3.90701783051876407669587130152, −2.93969727675367858733959102684, −1.57712576309046620530899965683, 0,
1.57712576309046620530899965683, 2.93969727675367858733959102684, 3.90701783051876407669587130152, 5.30256366122942019045428068996, 6.66814066786533764008775580892, 7.37289794031841749582743307472, 8.206422999833152166964222667455, 9.231213262047884386489035705174, 9.605284750774727919276580183024
