| L(s) = 1 | − 2-s − 3-s + 4-s + 0.561·5-s + 6-s − 0.561·7-s − 8-s + 9-s − 0.561·10-s − 2·11-s − 12-s + 3.12·13-s + 0.561·14-s − 0.561·15-s + 16-s − 6.56·17-s − 18-s + 2.56·19-s + 0.561·20-s + 0.561·21-s + 2·22-s − 23-s + 24-s − 4.68·25-s − 3.12·26-s − 27-s − 0.561·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.251·5-s + 0.408·6-s − 0.212·7-s − 0.353·8-s + 0.333·9-s − 0.177·10-s − 0.603·11-s − 0.288·12-s + 0.866·13-s + 0.150·14-s − 0.144·15-s + 0.250·16-s − 1.59·17-s − 0.235·18-s + 0.587·19-s + 0.125·20-s + 0.122·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 0.936·25-s − 0.612·26-s − 0.192·27-s − 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8822524612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8822524612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 - 5.12T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.567679896474643777049040896674, −7.67398400599443689656561522125, −7.06766181261656614305781480476, −6.13572466273971234404264805809, −5.82759956286418490819774086309, −4.72563430170498713625600803696, −3.87625869951588457828931646757, −2.71880214285323271311588368928, −1.82761372203279003886868453172, −0.60888474720614785080771244918,
0.60888474720614785080771244918, 1.82761372203279003886868453172, 2.71880214285323271311588368928, 3.87625869951588457828931646757, 4.72563430170498713625600803696, 5.82759956286418490819774086309, 6.13572466273971234404264805809, 7.06766181261656614305781480476, 7.67398400599443689656561522125, 8.567679896474643777049040896674
