L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s − 6·13-s − 2·17-s + 4·19-s + 21-s + 8·23-s − 27-s − 2·29-s − 4·33-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 49-s + 2·51-s − 6·53-s − 4·57-s − 4·59-s + 6·61-s − 63-s + 4·67-s − 8·69-s − 8·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.488·67-s − 0.963·69-s − 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.410499848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410499848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.46693619845362800767851977364, −7.12254479926148520605906354881, −6.48826800207420337319331935365, −5.74990561677519513932736543337, −4.90231700710465055623185649171, −4.50260005577866811106015668951, −3.45214410255695773525697920303, −2.72660319090534570392735403605, −1.63116136026629059069201971383, −0.61304936406779951585717776297,
0.61304936406779951585717776297, 1.63116136026629059069201971383, 2.72660319090534570392735403605, 3.45214410255695773525697920303, 4.50260005577866811106015668951, 4.90231700710465055623185649171, 5.74990561677519513932736543337, 6.48826800207420337319331935365, 7.12254479926148520605906354881, 7.46693619845362800767851977364