| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 15.1·5-s + 6·6-s + 0.939·7-s + 8·8-s + 9·9-s + 30.3·10-s + 50.0·11-s + 12·12-s − 30.5·13-s + 1.87·14-s + 45.5·15-s + 16·16-s − 36.9·17-s + 18·18-s + 60.7·20-s + 2.81·21-s + 100.·22-s − 24.4·23-s + 24·24-s + 105.·25-s − 61.1·26-s + 27·27-s + 3.75·28-s + 142.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.35·5-s + 0.408·6-s + 0.0507·7-s + 0.353·8-s + 0.333·9-s + 0.960·10-s + 1.37·11-s + 0.288·12-s − 0.652·13-s + 0.0358·14-s + 0.784·15-s + 0.250·16-s − 0.527·17-s + 0.235·18-s + 0.679·20-s + 0.0292·21-s + 0.970·22-s − 0.221·23-s + 0.204·24-s + 0.845·25-s − 0.461·26-s + 0.192·27-s + 0.0253·28-s + 0.915·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.901996555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.901996555\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 15.1T + 125T^{2} \) |
| 7 | \( 1 - 0.939T + 343T^{2} \) |
| 11 | \( 1 - 50.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 24.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 85.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 131.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 708.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 38.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 582.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 46.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 894.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
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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
−8.804967152373775974376571831630, −7.964697137047797550070274177573, −6.72266415720039435318482236029, −6.53250651784078840545673959030, −5.52362788923875494053823663682, −4.66708949861341130409176533092, −3.85868845379214787828254760545, −2.73856172639100743686855279152, −2.05548758666177748197139091116, −1.12185948817988086961206421272,
1.12185948817988086961206421272, 2.05548758666177748197139091116, 2.73856172639100743686855279152, 3.85868845379214787828254760545, 4.66708949861341130409176533092, 5.52362788923875494053823663682, 6.53250651784078840545673959030, 6.72266415720039435318482236029, 7.964697137047797550070274177573, 8.804967152373775974376571831630
