| L(s) = 1 | − 9.22e4·2-s − 4.30e7·3-s − 7.79e7·4-s − 3.77e10·5-s + 3.97e12·6-s − 2.13e13·7-s + 7.99e14·8-s + 1.85e15·9-s + 3.48e15·10-s + 1.25e17·11-s + 3.35e15·12-s + 2.98e18·13-s + 1.97e18·14-s + 1.62e18·15-s − 7.31e19·16-s − 1.73e20·17-s − 1.70e20·18-s − 4.45e19·19-s + 2.94e18·20-s + 9.19e20·21-s − 1.16e22·22-s − 3.44e22·23-s − 3.44e22·24-s − 1.14e23·25-s − 2.75e23·26-s − 7.97e22·27-s + 1.66e21·28-s + ⋯ |
| L(s) = 1 | − 0.995·2-s − 0.577·3-s − 0.00907·4-s − 0.110·5-s + 0.574·6-s − 0.243·7-s + 1.00·8-s + 0.333·9-s + 0.110·10-s + 0.825·11-s + 0.00524·12-s + 1.24·13-s + 0.241·14-s + 0.0638·15-s − 0.990·16-s − 0.863·17-s − 0.331·18-s − 0.0354·19-s + 0.00100·20-s + 0.140·21-s − 0.821·22-s − 1.17·23-s − 0.579·24-s − 0.987·25-s − 1.23·26-s − 0.192·27-s + 0.00220·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.30e7T \) |
| good | 2 | \( 1 + 9.22e4T + 8.58e9T^{2} \) |
| 5 | \( 1 + 3.77e10T + 1.16e23T^{2} \) |
| 7 | \( 1 + 2.13e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.25e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.98e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 1.73e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 4.45e19T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.44e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 3.07e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.39e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.27e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 6.60e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 9.67e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 1.84e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 1.46e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.77e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.00e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.10e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 4.12e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 5.52e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 9.45e29T + 4.18e62T^{2} \) |
| 83 | \( 1 + 7.81e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 7.99e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.07e33T + 3.65e65T^{2} \) |
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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
−17.31748174516579445758980294357, −15.94348163227884232240622695556, −13.53385783499617849588113184065, −11.44877366421240981001886355255, −9.873132899281669554709942652041, −8.337221878198501816246778948455, −6.39807142629599475039708588895, −4.18594927259881472595598639142, −1.41779393831383572931708372171, 0,
1.41779393831383572931708372171, 4.18594927259881472595598639142, 6.39807142629599475039708588895, 8.337221878198501816246778948455, 9.873132899281669554709942652041, 11.44877366421240981001886355255, 13.53385783499617849588113184065, 15.94348163227884232240622695556, 17.31748174516579445758980294357
