L(s) = 1 | + (1.50 + 1.09i)3-s + (−0.809 + 0.587i)4-s + (0.541 − 1.66i)5-s + (−1.17 + 0.856i)7-s + (0.759 + 2.33i)9-s + (0.968 − 0.248i)11-s − 1.85·12-s + (0.331 + 1.01i)13-s + (2.63 − 1.91i)15-s + (0.309 − 0.951i)16-s + (0.541 + 1.66i)20-s − 2.71·21-s + (−1.67 − 1.21i)25-s + (−0.837 + 2.57i)27-s + (0.450 − 1.38i)28-s + ⋯ |
L(s) = 1 | + (1.50 + 1.09i)3-s + (−0.809 + 0.587i)4-s + (0.541 − 1.66i)5-s + (−1.17 + 0.856i)7-s + (0.759 + 2.33i)9-s + (0.968 − 0.248i)11-s − 1.85·12-s + (0.331 + 1.01i)13-s + (2.63 − 1.91i)15-s + (0.309 − 0.951i)16-s + (0.541 + 1.66i)20-s − 2.71·21-s + (−1.67 − 1.21i)25-s + (−0.837 + 2.57i)27-s + (0.450 − 1.38i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526971278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526971278\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.968 + 0.248i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.98T + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−9.506189150252004917652348457947, −9.176045854378435001212632230568, −8.538935860702995802663917376395, −8.164687320162107103046232722163, −6.58318961177792916694845937564, −5.36928495117008190412792716123, −4.59051035927310234079420535654, −3.89504257867996471024099131455, −3.14851880120488134429331903075, −1.86122459171333006384042479221,
1.25191162432772337907878948151, 2.55272605768023836687961193669, 3.40234619243797604288240809360, 3.91452509468985041480615621119, 5.91679963132776309889057338376, 6.55512206989556622610799130480, 7.07347817931145150721025809375, 7.85175125151316321933603824039, 8.899123977006944040242698844294, 9.566585568040747361012529651331