| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s − 11-s + 3·13-s − 3·15-s − 2·17-s − 3·19-s − 21-s − 4·23-s + 4·25-s + 27-s − 9·29-s − 2·31-s − 33-s + 3·35-s − 11·37-s + 3·39-s − 4·41-s − 4·43-s − 3·45-s − 3·47-s + 49-s − 2·51-s − 4·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s − 0.485·17-s − 0.688·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.174·33-s + 0.507·35-s − 1.80·37-s + 0.480·39-s − 0.624·41-s − 0.609·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.550209289339973999946918010751, −8.607584234319820279843149378887, −8.112770171371127759402374804737, −7.24264125802517025269438845065, −6.40761147031881701581107354593, −5.11675518536581268064805822554, −3.85808269239208963151847638821, −3.53426831848528683359566672044, −2.00543267362711116052566627685, 0,
2.00543267362711116052566627685, 3.53426831848528683359566672044, 3.85808269239208963151847638821, 5.11675518536581268064805822554, 6.40761147031881701581107354593, 7.24264125802517025269438845065, 8.112770171371127759402374804737, 8.607584234319820279843149378887, 9.550209289339973999946918010751
