| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 4·13-s + 2·15-s − 2·17-s − 4·19-s + 2·21-s − 4·23-s − 25-s + 27-s + 6·29-s + 2·31-s + 4·35-s + 8·37-s + 4·39-s − 2·41-s − 4·43-s + 2·45-s − 12·47-s − 3·49-s − 2·51-s + 6·53-s − 4·57-s + 4·59-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.676·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.75·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.366233936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366233936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.21776117870948491031897964139, −9.497476761258686531860854084973, −8.420702100846206882694460804856, −8.137356992201587112320496664394, −6.68785915312412950432661260762, −6.03168549170604009889878923550, −4.84641162437662246944769643988, −3.87678607648250663694375372790, −2.47902910741521785662962015803, −1.51663712403377833666587682944,
1.51663712403377833666587682944, 2.47902910741521785662962015803, 3.87678607648250663694375372790, 4.84641162437662246944769643988, 6.03168549170604009889878923550, 6.68785915312412950432661260762, 8.137356992201587112320496664394, 8.420702100846206882694460804856, 9.497476761258686531860854084973, 10.21776117870948491031897964139
