| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s + 3·11-s − 12-s − 2·13-s + 3·14-s + 15-s + 16-s − 4·17-s − 18-s − 3·19-s − 20-s + 3·21-s − 3·22-s + 5·23-s + 24-s + 25-s + 2·26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.688·19-s − 0.223·20-s + 0.654·21-s − 0.639·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6265047031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6265047031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03511220966427259395599233711, −9.229056355682263950262848905731, −8.623352409545617599009062450677, −7.35369985427599614219903537999, −6.71829629795649733134253151301, −6.10717089690504172327523562120, −4.73913809507066654654368393960, −3.70057664213916508271525593892, −2.44172357185384456645633702554, −0.69654490443051897026111019883,
0.69654490443051897026111019883, 2.44172357185384456645633702554, 3.70057664213916508271525593892, 4.73913809507066654654368393960, 6.10717089690504172327523562120, 6.71829629795649733134253151301, 7.35369985427599614219903537999, 8.623352409545617599009062450677, 9.229056355682263950262848905731, 10.03511220966427259395599233711
