| L(s) = 1 | + 5.65i·2-s − 32.0·4-s − 24.8i·5-s − 6.19·7-s − 181. i·8-s + 140.·10-s + 130. i·11-s − 922.·13-s − 35.0i·14-s + 1.02e3·16-s − 3.38e3i·17-s + 5.40e3·19-s + 794. i·20-s − 740.·22-s + 2.36e3i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.198i·5-s − 0.0180·7-s − 0.353i·8-s + 0.140·10-s + 0.0983i·11-s − 0.419·13-s − 0.0127i·14-s + 0.250·16-s − 0.688i·17-s + 0.787·19-s + 0.0992i·20-s − 0.0695·22-s + 0.194i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.666957347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666957347\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 24.8iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 6.19T + 1.17e5T^{2} \) |
| 11 | \( 1 - 130. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 922.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.40e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.36e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.49e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 4.66e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 6.91e3T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.38e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.23e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.60e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.32e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.90e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.20e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.29e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 6.28e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.87e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.71e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.35e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.01e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 4.18e5T + 8.32e11T^{2} \) |
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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
−12.17273286561426335234510804519, −10.97009340293895737513523714509, −9.721681722110058819379387556930, −8.909182436333743728544929509819, −7.69911966681077427095742372652, −6.83792085026262081554111260614, −5.51357133450215452671542029177, −4.55643415361811843409152634271, −2.97299123776464371834841582617, −1.02765213242165244982656942075,
0.60825224092598202943167468638, 2.15065574576836204074426493832, 3.41748306930860739481483671933, 4.70088089819609813924858913189, 6.01337324632194373119299402676, 7.40765341690456207880144272247, 8.564081670731083971015383943257, 9.662541171565286228910613855795, 10.52959532295513050359998844929, 11.50318721116501458078081881933
