| L(s) = 1 | + 10.2·2-s + 19.9·3-s + 73.7·4-s + 39.6·5-s + 205.·6-s + 49·7-s + 429.·8-s + 155.·9-s + 407.·10-s − 121·11-s + 1.47e3·12-s − 169·13-s + 503.·14-s + 790.·15-s + 2.05e3·16-s − 420.·17-s + 1.59e3·18-s + 2.64e3·19-s + 2.92e3·20-s + 977.·21-s − 1.24e3·22-s + 2.91e3·23-s + 8.57e3·24-s − 1.55e3·25-s − 1.73e3·26-s − 1.75e3·27-s + 3.61e3·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 1.28·3-s + 2.30·4-s + 0.708·5-s + 2.32·6-s + 0.377·7-s + 2.37·8-s + 0.638·9-s + 1.28·10-s − 0.301·11-s + 2.95·12-s − 0.277·13-s + 0.687·14-s + 0.907·15-s + 2.01·16-s − 0.352·17-s + 1.16·18-s + 1.67·19-s + 1.63·20-s + 0.483·21-s − 0.548·22-s + 1.14·23-s + 3.03·24-s − 0.497·25-s − 0.504·26-s − 0.462·27-s + 0.871·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(15.25908698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.25908698\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 49T \) |
| 11 | \( 1 + 121T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 3 | \( 1 - 19.9T + 243T^{2} \) |
| 5 | \( 1 - 39.6T + 3.12e3T^{2} \) |
| 17 | \( 1 + 420.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.52e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.29e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.33e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.26e4T + 8.58e9T^{2} \) |
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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
−9.307685539928583558348068343869, −8.188030818064389205166694927366, −7.40284642642297818018079531225, −6.55697697928652992997221571919, −5.47373478742910921952031272660, −4.92437052316926862801682644107, −3.81668597641378915076723650151, −2.96097573865332717408635464471, −2.40703864499070146563514782901, −1.40258758382358876272489075084,
1.40258758382358876272489075084, 2.40703864499070146563514782901, 2.96097573865332717408635464471, 3.81668597641378915076723650151, 4.92437052316926862801682644107, 5.47373478742910921952031272660, 6.55697697928652992997221571919, 7.40284642642297818018079531225, 8.188030818064389205166694927366, 9.307685539928583558348068343869
