| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 13-s − 15-s − 2·17-s + 5·19-s + 21-s + 2·23-s + 25-s − 27-s + 8·29-s − 7·31-s − 35-s − 7·37-s − 39-s − 6·41-s − 8·43-s + 45-s + 8·47-s − 6·49-s + 2·51-s − 6·53-s − 5·57-s + 5·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.485·17-s + 1.14·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s − 1.25·31-s − 0.169·35-s − 1.15·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 1.16·47-s − 6/7·49-s + 0.280·51-s − 0.824·53-s − 0.662·57-s + 0.640·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.33933010694697, −12.87791134489268, −12.35147278729333, −12.01624146844985, −11.45023437069029, −10.98393711345586, −10.58489842732203, −9.984958071971446, −9.719212313609761, −9.126155514334975, −8.638700170035889, −8.178128694603151, −7.462588548243967, −6.894053902770338, −6.685801606271238, −6.070257844818312, −5.523630665287799, −5.039636787659023, −4.715238171569849, −3.841085353738761, −3.361191508047189, −2.860879298995333, −2.027559843917452, −1.488508581681941, −0.7981314255056598, 0,
0.7981314255056598, 1.488508581681941, 2.027559843917452, 2.860879298995333, 3.361191508047189, 3.841085353738761, 4.715238171569849, 5.039636787659023, 5.523630665287799, 6.070257844818312, 6.685801606271238, 6.894053902770338, 7.462588548243967, 8.178128694603151, 8.638700170035889, 9.126155514334975, 9.719212313609761, 9.984958071971446, 10.58489842732203, 10.98393711345586, 11.45023437069029, 12.01624146844985, 12.35147278729333, 12.87791134489268, 13.33933010694697
