L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s + 4·21-s − 8·23-s + 25-s + 27-s + 6·29-s − 4·31-s + 4·33-s − 4·35-s − 10·37-s − 2·39-s + 2·41-s − 12·43-s − 45-s + 9·49-s − 2·51-s + 6·53-s − 4·55-s + 10·61-s + 4·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 0.149·45-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s + 1.28·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.59852915701011, −12.14786612918307, −11.97416968342448, −11.49400743720051, −11.05775135268561, −10.50353883179607, −10.08177131460050, −9.560540241400179, −9.030100814365349, −8.555645281626559, −8.155775301331270, −7.992730975199585, −7.236195735250704, −6.830325266111859, −6.505503308697774, −5.645317445721167, −5.183346297500556, −4.677823774226875, −4.212074960395446, −3.785818951558692, −3.332749938005955, −2.442029418990971, −1.965690308421906, −1.598921548700221, −0.8905587925593588, 0,
0.8905587925593588, 1.598921548700221, 1.965690308421906, 2.442029418990971, 3.332749938005955, 3.785818951558692, 4.212074960395446, 4.677823774226875, 5.183346297500556, 5.645317445721167, 6.505503308697774, 6.830325266111859, 7.236195735250704, 7.992730975199585, 8.155775301331270, 8.555645281626559, 9.030100814365349, 9.560540241400179, 10.08177131460050, 10.50353883179607, 11.05775135268561, 11.49400743720051, 11.97416968342448, 12.14786612918307, 12.59852915701011