| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 0.660·5-s − 6·6-s + 1.36·7-s + 8·8-s + 9·9-s − 1.32·10-s + 34.0·11-s − 12·12-s − 38.7·13-s + 2.72·14-s + 1.98·15-s + 16·16-s + 18·18-s − 19.3·19-s − 2.64·20-s − 4.08·21-s + 68.1·22-s − 121.·23-s − 24·24-s − 124.·25-s − 77.5·26-s − 27·27-s + 5.44·28-s + 51.7·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0590·5-s − 0.408·6-s + 0.0735·7-s + 0.353·8-s + 0.333·9-s − 0.0417·10-s + 0.933·11-s − 0.288·12-s − 0.827·13-s + 0.0519·14-s + 0.0341·15-s + 0.250·16-s + 0.235·18-s − 0.233·19-s − 0.0295·20-s − 0.0424·21-s + 0.660·22-s − 1.09·23-s − 0.204·24-s − 0.996·25-s − 0.585·26-s − 0.192·27-s + 0.0367·28-s + 0.331·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 0.660T + 125T^{2} \) |
| 7 | \( 1 - 1.36T + 343T^{2} \) |
| 11 | \( 1 - 34.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 19.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 51.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 78.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 32.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 591.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 127.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 71.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 203.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 920.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 295.T + 9.12e5T^{2} \) |
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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
−8.405604950925971271105783958988, −7.62141897662195548723138223579, −6.69923983731111102318145185926, −6.15036599113768432863346237174, −5.25122479166638450961095202500, −4.41118945120876055510552839443, −3.72696279069502685647910759941, −2.46127555117531804633672753116, −1.41813374776463548884666459799, 0,
1.41813374776463548884666459799, 2.46127555117531804633672753116, 3.72696279069502685647910759941, 4.41118945120876055510552839443, 5.25122479166638450961095202500, 6.15036599113768432863346237174, 6.69923983731111102318145185926, 7.62141897662195548723138223579, 8.405604950925971271105783958988
