L(s) = 1 | + 1.82·2-s − 3-s + 1.33·4-s + 5-s − 1.82·6-s + 1.74·7-s − 1.20·8-s + 9-s + 1.82·10-s − 1.33·12-s − 5.35·13-s + 3.19·14-s − 15-s − 4.88·16-s + 7.33·17-s + 1.82·18-s + 6.01·19-s + 1.33·20-s − 1.74·21-s + 8.35·23-s + 1.20·24-s + 25-s − 9.78·26-s − 27-s + 2.33·28-s + 5.53·29-s − 1.82·30-s + ⋯ |
L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.669·4-s + 0.447·5-s − 0.745·6-s + 0.660·7-s − 0.427·8-s + 0.333·9-s + 0.577·10-s − 0.386·12-s − 1.48·13-s + 0.853·14-s − 0.258·15-s − 1.22·16-s + 1.77·17-s + 0.430·18-s + 1.37·19-s + 0.299·20-s − 0.381·21-s + 1.74·23-s + 0.246·24-s + 0.200·25-s − 1.91·26-s − 0.192·27-s + 0.441·28-s + 1.02·29-s − 0.333·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.168362693\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.168362693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 - 7.33T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 - 5.53T + 29T^{2} \) |
| 31 | \( 1 + 0.0213T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 - 0.350T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
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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
−9.568974856175062767214074660103, −8.379879814432190371918242588343, −7.33145263197254859418214424367, −6.77761761355088518238644026131, −5.50993311703765274829371785031, −5.27035060394213942450180266277, −4.66788415761887999869695955246, −3.41617368941739495538208511902, −2.61365091118733667565863488296, −1.10625156101449549334201025824,
1.10625156101449549334201025824, 2.61365091118733667565863488296, 3.41617368941739495538208511902, 4.66788415761887999869695955246, 5.27035060394213942450180266277, 5.50993311703765274829371785031, 6.77761761355088518238644026131, 7.33145263197254859418214424367, 8.379879814432190371918242588343, 9.568974856175062767214074660103