| L(s) = 1 | + 27.1·2-s − 167.·3-s + 223.·4-s − 1.05e3·5-s − 4.53e3·6-s + 2.40e3·7-s − 7.81e3·8-s + 8.29e3·9-s − 2.87e4·10-s − 2.76e4·11-s − 3.74e4·12-s − 2.85e4·13-s + 6.51e4·14-s + 1.77e5·15-s − 3.26e5·16-s − 5.12e5·17-s + 2.25e5·18-s + 6.91e5·19-s − 2.37e5·20-s − 4.01e5·21-s − 7.48e5·22-s + 1.39e6·23-s + 1.30e6·24-s − 8.31e5·25-s − 7.74e5·26-s + 1.90e6·27-s + 5.37e5·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 1.19·3-s + 0.437·4-s − 0.757·5-s − 1.42·6-s + 0.377·7-s − 0.674·8-s + 0.421·9-s − 0.908·10-s − 0.568·11-s − 0.521·12-s − 0.277·13-s + 0.453·14-s + 0.903·15-s − 1.24·16-s − 1.48·17-s + 0.505·18-s + 1.21·19-s − 0.331·20-s − 0.450·21-s − 0.681·22-s + 1.03·23-s + 0.804·24-s − 0.425·25-s − 0.332·26-s + 0.689·27-s + 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.318272002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318272002\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 27.1T + 512T^{2} \) |
| 3 | \( 1 + 167.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.05e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.76e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.12e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.39e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.62e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.19e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.35e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.95e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.42e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.69e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.41e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.61e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.80e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.43e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.08e9T + 7.60e17T^{2} \) |
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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
−12.10439683506384678591383494156, −11.59169203987746562353872130674, −10.65602880440020385272837712497, −8.912417574715548984073758018286, −7.36069591428687583099516648712, −6.11777715911967163331478299276, −5.06806848674876462905187211569, −4.35275132365358843644671211887, −2.81014815719410616949180601715, −0.56222213535582695671047548599,
0.56222213535582695671047548599, 2.81014815719410616949180601715, 4.35275132365358843644671211887, 5.06806848674876462905187211569, 6.11777715911967163331478299276, 7.36069591428687583099516648712, 8.912417574715548984073758018286, 10.65602880440020385272837712497, 11.59169203987746562353872130674, 12.10439683506384678591383494156
