L(s) = 1 | − 35.6·2-s + 190.·3-s + 759.·4-s − 886.·5-s − 6.79e3·6-s − 3.87e3·7-s − 8.81e3·8-s + 1.66e4·9-s + 3.15e4·10-s + 2.60e4·11-s + 1.44e5·12-s − 6.57e3·13-s + 1.38e5·14-s − 1.68e5·15-s − 7.43e4·16-s − 1.49e5·17-s − 5.92e5·18-s − 1.43e5·19-s − 6.72e5·20-s − 7.38e5·21-s − 9.29e5·22-s − 3.28e5·23-s − 1.67e6·24-s − 1.16e6·25-s + 2.34e5·26-s − 5.84e5·27-s − 2.94e6·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s + 1.35·3-s + 1.48·4-s − 0.634·5-s − 2.13·6-s − 0.610·7-s − 0.761·8-s + 0.844·9-s + 0.999·10-s + 0.536·11-s + 2.01·12-s − 0.0638·13-s + 0.962·14-s − 0.861·15-s − 0.283·16-s − 0.433·17-s − 1.33·18-s − 0.252·19-s − 0.940·20-s − 0.829·21-s − 0.845·22-s − 0.244·23-s − 1.03·24-s − 0.597·25-s + 0.100·26-s − 0.211·27-s − 0.905·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + 7.07e5T \) |
good | 2 | \( 1 + 35.6T + 512T^{2} \) |
| 3 | \( 1 - 190.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 886.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.57e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.49e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.28e5T + 1.80e12T^{2} \) |
| 31 | \( 1 + 8.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.11e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.46e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.85e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.55e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.48e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.60e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.93e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.45e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.17e9T + 7.60e17T^{2} \) |
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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
−14.75461550327117772772490883053, −13.27679443066149296310812981234, −11.51174718435144581211680314761, −9.957436229195675358423885668429, −8.992383058377539193453741041621, −8.112407328406468554420558009966, −6.94932059545327105533932723246, −3.62479041217308518237313867451, −1.95835826654431768371396625325, 0,
1.95835826654431768371396625325, 3.62479041217308518237313867451, 6.94932059545327105533932723246, 8.112407328406468554420558009966, 8.992383058377539193453741041621, 9.957436229195675358423885668429, 11.51174718435144581211680314761, 13.27679443066149296310812981234, 14.75461550327117772772490883053